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All Textbook Solutions for Precalculus
For Exercises 29-32, solve ABC subject to the given conditions if possible. Round the lengths of the sides and measures of the angles (in degrees) to 1 decimal place if necessary. a=17.3,b=24.2,C=6332REFor Exercises 33-34, find the measures of the angles within the triangle formed by the given points. Round to the nearest tenth of a degree. A1,0,B5,2,C2,534RE35REIn Exercises 35-36, find the area of the triangle with sides of the given lengths. Round to the nearest tenth of a square unit. a=47mm,b=32mm,c=60mmTwo straight jogging paths begin at a kiosk in a park, and the angle between the paths is 73 . Two runners leave the kiosk at the same time each taking one of the paths. One person runs at a rate of 5mph , the other at a rate of 6.2mph . After 30min , how far apart are the runners? Round to the nearest tenth of a mile.Given points A6,2 and B1,4 on the coordinate plane, find the measure of angleAOB , where point O is at the origin. Round to the nearest tenth of a degree.39RETwo planes take off from an airport at the same time. The first plane averages 480mph and flies on a bearing of N20E . The second plane averages 360mph and heads out at a bearing ofN35W . a. After 1.5hr , how far apart are the two planes? Round to the nearest tenth of a mile. b. Find the bearing from the first plane to the second plane. Round to the nearest tenth of a degree.Two sides of a triangular plot of land measure 510ft and 237ft and the angle between them is 78.1 a. If the land is 5500 per acre, what is the cost for this plot? Round to the nearest dollar and use the fact that 1 acre is4840yd2 . b. If materials for a chain-link fence cost 4.20 per linear foot, what is the cost to completely enclose the property? Round to the nearest dollar.42REFor Exercises 43-46, suppose that an object moves in simple harmonic motion with displacement d (in centimeters) at time t (in seconds). Determine the a. Amplitude b. Period c. Frequency d. Phase Shift d=8cos6t44REFor Exercises 43-46, suppose that an object moves in simple harmonic motion with displacement d (in centimeters) at time t (in seconds). Determine the a. Amplitude b. Period c. Frequency d. Phase Shift d2sint346RE47REFor Exercises 47-48, a block is attached to a horizontal spring. At time t=0 , if the block is moving to the left with an initial displacement of 0ft , a maximum displacement of 13 and a frequency of3Hz , find a model for the displacement d as a function of time t .The angular displacement (in radians) for a simple pendulum is given by =0.175sint . a. Determine the period of the pendulum. b. How many swings are completed in 1sec ? c. To the nearest degree, what is the maximum displacement of the pendulum? d. The length L of a pendulum is related to the period T by the equation T2Lg , whereg is the acceleration due to gravity. Find the length of the pendulum to the nearest meter g=9.8m/sec2 .A Ferris wheel is 200ft in diameter with its lowest point 3ft off the ground. Once all the passengers have been loaded, the wheel makes one full rotation counterclockwise in 3min . Suppose that a couple is seated at the lowest point on the wheel and are the last passengersr to be loaded. a. Write a model representing the couple's horizontal position x (in feet) relative to the center of the Ferris wheel, t minutes after the ride starts. b. Write a model representing the couple's height y (in feet) above ground level, t minutes after the ride starts. c. Give the coordinates of the couple's position 2min into the ride and describe the position.For Exercises 51-52, use a graphing utility to graph the function and bounding curves for t0 . d=4e0.03tcos1.4tFor Exercises 51-52, use a graphing utility to graph the function and bounding curves for t0 . d=6e1.3tsin5tFor Exercises 1-8, solve ABC subject to the given conditions. Round the lengths of sides and measures of the angles to 1 decimal place if necessary. A=62,C=90,a=17For Exercises 1-8, solve ABC subject to the given conditions. Round the lengths of sides and measures of the angles to 1 decimal place if necessary. a=17.6,b=20,c=12.8For Exercises 1-8, solve ABC subject to the given conditions. Round the lengths of sides and measures of the angles to 1 decimal place if necessary. B=97.3,C=12.7,b=77.1For Exercises 1-8, solve ABC subject to the given conditions. Round the lengths of sides and measures of the angles to 1 decimal place if necessary. B=14.8,C=90,a=2135T6TFor Exercises 1-8, solve ABC subject to the given conditions. Round the lengths of sides and measures of the angles to 1 decimal place if necessary. B=13,a=7,b=28T9T10TFor Exercises 9-12, find the area of ABC with the given measurements. Round to 1 decimal place. a=28m,b=46m,c=23m12TA ship travels west at 25 knots. At 7:30P.M. , the captain sees a floating become due south of the ship. By 9:15P.M. , the captain sights the become at a bearing of S32E . What was the distance from the ship to the become at the time of the first sighting? Round to the nearest nautical mile.Over the course of a day, as the angle of inclination of the Sun decreases from 64 to 49 , the length of the shadow of a building increases by 60ft How tall is the building? Round to the nearest foot.To determine the height of a tower, a surveyor notes that the angle of elevation to the top of the tower is 48 when measured at a point on the ground 45ft from the base of the tower. To the nearest tenth of a foot, how tall is the tower?In addition to the light-year to quantify large distances in outer space, astronomers often use two other units of measurement. • the astronomical unit AU , which is the average distance between the Earth and Sun1AU93,000,000mi . • the parsec (pc), which is the distance from the Earth to an object having a parallax angle of 1 (1 second of arc). a. To the nearest whole unit how many astronomical units are in 1 parsec? b. How many light-years are in 1 parsec? Round to 2 decimal places. (Hint 1 light-year =5.8781012mi ).A triangular shade sail has sides of length 123 and 156 , and the angle between them is7318 . a. If the cost of the sail material is 8.75/ft2 , how much would this sail cost? Round to the nearest dollar. b. If the cost of wrapping the edges of the sail with a decorative border in a contrasting color is125/yd , what is the added cost to the sail? Round to the nearest dollar.The distance between Los Angeles, California, and Salt Lake City, Utah, on a map is 9.5in . The distance from Salt Lake City to Albuquerque, New Mexico, is 8.0in. , and the distance from Albuquerque to Los Angeles is 11.0in . If the bearing from Salt Lake City to Los Angeles is S34W , find the bearing from Albuquerque to Los Angeles. Round to the nearest tenth of a degree.The connector rod from the piston to the crankshaft in a certain engine is 8.5in . The radius of the crank circle is 3 in. If the angle made by the connector rod with the vertical at the wrist pin P is 12 , how far is the wrist pin from the center C of the crankshaft? Round to the nearest tenth of an inch.20T21TThe displacement of an object attached to a spring that moves in simple harmonic motion is given by d=1.8cos12t , where the displacement d is in centimeters at time t in seconds. Determine the amplitude and frequency for the motion of the object.23TA block hangs on a spring attached to the ceiling and is pulled down 8 in. below its equilibrium position. After release, the block makes one complete up-and-down cycle in 1.2sec and follows simple harmonic motion. a. Write a function to model the displacement d (in inches) as a function of the time d (in seconds) after release. Assume that a displacement above the equilibrium position is positive. b. Find the displacement of the block and direction of movement at t=0.3sec .The angular displacement (in radians) for a simple pendulum is given by.=0.14sin2t . a. Determine the period of the pendulum. b. How many swings are completed in 1sec ? c. To the nearest degree, what is the maximum displacement of the pendulum? d. The length L of a pendulum is related to grain the T by the equation T=2LR , where, g is the acceleration due to gravity. Find the length of the pendulum to the nearest foot g=32ft/sec2 .26TUse a graphing utility to graph the function d=4e0.25esin1.3t and bounding curves for t0 .Write the equation of the line in slope-intercept form that passes through the point 3,12 with slope m=23 .2CRE3CREWrite a polynomial fx of degree 5 , with integer coefficients, and with zeros 2 (multiplicity 2 ), 6 , and 2i .Write equations for the asymptotes for the graph of fx=2x2+5x3x8Write equations for the asymptotes for the graph of fxsec2x on the interval 2,27CRE8CREWrite an equation for a tangent function with a period of 13 and a phase shift of 43 .A motorist drives on a toll road to and from work each day and pays 4.25 in tolls one-way. a. Write a model for the costCin for tolls in terms of the number of working days, x . b. The department of transportation offers a prepaid toll pass for 150 a month. How many working days are required for the motorist to save money by buying the pass?a. Write a variation model using k as the constant of variation if Z is directly proportional to the cube of A and inversely proportional to the square root of 8 . b. If Z=100 when A=2 and B=25 , what is the constant of proportionality k ?a. Write an equation representing the fact that the product of two consecutive odd integers is 323 . b. Solve the equation from part (a) to find the two integers.13CREThe cost to buy tickets to a concert is $50 per ticket. a. Write a function that represents the cost Cxin for x tickets to the concert. b. There is a sales tax of 6 and a processing fee of 5 for a group of tickets. Write a function that represents the total costa for a dollars spent on tickets. c. Find TCX . d. Find TC10 and interpret its meaning in the context of this problem.Suppose that the population of a country in the year 2000 was 19.0 million and grew to 22.6 million in 2010 . Write a model of the form ptp0e, where pt is the population in millions, t years after the year 2000 . Round the growth rate to 5 decimal places.16CRE17CREFind the exact value of cos200cos50+sin200sin5019CREFor Exercises 19-20, solve ABC subject to the given conditions. Round the lengths of sides and measures of the angles to 1 decimal place if necessary. a=12.8,b=20,B=98.7Solve the right triangle. Round the lengths of the sides to 1 decimal place.A 40-ft ladder on a fire truck leans against a burning building such that the top of the ladder makes an angle of 20 with the building. If the bottom of the ladder is anchored to the top of the fire truck 6ft off the ground, how far above ground does the ladder reach? Round to the nearest tenth of a foot.From an eye level of 6ft , an observer standing 30ft away from a building sights the bottom of a window at an angle of elevation of6.1 . He sights the top of the window at an angle of elevation of 20.1 . Find the vertical dimension of the window. Round to the nearest tenth of a foot.A straight hiking trail is 2500ft long with a gain in altitude of 310ft . To the nearest degree, what is the average angle of ascent?5SP6SPA plane leaves an airport heading S50E at 450mph . After 2hr , the plane makes a 90 clockwise turn to a new bearing of S40W . If the plane travels this new course for 12hr , a. Find the plane's distance from the airport to the nearest mile. b. To the nearest degree, find the bearing from the airport to the plane.For Exercise 1-2, refer triangle to ABC . If the lengths of sides a and b are known, the inverse or function can be used directly to find the value of angle A .For Exercise 1-2 refer triangle to ABC . The measure of angle B and the length of side c can be used along with which trigonometric functions to find the length of side a ?3PEThe acute angle used to express the bearing between two points is measured relative to the (north-south or east-west) line.For exercise 5-12 given right triangle ABC , determine if the expression is true or false. cotB=abFor exercise 5-12 given right triangle ABC , determine if the expression is true or false. sin1ab=AFor exercise 5-12 given right triangle ABC , determine if the expression is true or false. cscB=caFor exercise 5-12 given right triangle ABC , determine if the expression is true or false. secA=cbFor exercise 5-12 given right triangle ABC , determine if the expression is true or false. cos1ac=BFor exercise 5-12 given right triangle ABC , determine if the expression is true or false. A+B=9011PE12PEFor Exercises 13-18, solve the right triangle for the unknown sides and angles. Round values to 1 decimal place if necessary. B=24,c=18For Exercises 13-18, solve the right triangle for the unknown sides and angles. Round values to 1 decimal place if necessary. A=28,c=15For Exercises 13-18, solve the right triangle for the unknown sides and angles. Round values to 1 decimal place if necessary. a=10,b=7For Exercises 13-18, solve the right triangle for the unknown sides and angles. Round values to 1 decimal place if necessary. a=2,b=8.6For Exercises 13-18, solve the right triangle for the unknown sides and angles. Round values to 1 decimal place if necessary. A=38.2,b=17.8For Exercises 13-18, solve the right triangle for the unknown sides and angles. Round values to 1 decimal place if necessary. A=25.6,a=34A boat is anchored off Elliot Key, Florida. From the bow of the boat, 36ft of anchor line is out with 4ft of line above the water. The angle that the line makes with the water line is 22 a. How deep is the water? Round to the nearest foot. b. What is the horizontal distance between the anchor and the bow of the boat? Round to the nearest foot.A swimming pool has a depth of 4ft at the shallow end and 8ft at the deep end. The bottom of the pool slopes downward at an angle of 5.7 . How long is the pool? Round to the nearest foot.An airplane climbs at an angle of 11 at an average speed of 420mph . How long will it take for the plane to reach its cruising altitude of 6.5mi ? Round to the nearest minute.An airplane begins its descent with an average speed of 240mph at an angle of depression of 4 . How much altitude will the plane lose in 5min . Round to the nearest tenth of a mile.A railroad bridge over New Scotland Road in Slingerlands, New York, has a low clearance for trucks. An engineer standing 20ft away measures a 15.4 angle of elevation from her eye level of 5.5ft to the bottom of the bridge. If the road is flat between the engineer and the bridge, how high over the roadway is the bottom of the bridge? Round to the nearest inch.At 11:30AM. the angle of elevation of the sun for Washington, D.C., is 55.7 . If the height of the Washington monument is approximately 555ft , what is the length of the shadow it will cast at that time? Round to the nearest foot.25PERoberto has plans to build a tree house in a large oak tree in his backyard. He takes measurements from the ground and uses trigonometry to estimate how tall the tree house can be. Twenty feet from the base of the tree, Roberto measures a 26.6 angle of elevation to the proposed bottom of the tree house and a 42 angle of elevation to the proposed top of the tree house. If Roberto is 6ft tall, will there be enough room for him to stand inside the tree house?A police officer hiding between two bushes 50 ft from a straight highway sights two points A , and B . The angle from the police car to A is 62 and the angle to point B is 72 . a. Find the distance between A and B . Round to the nearest foot. b. Suppose that a motorist takes 2.7 sec to pass from A to B . Using the rounded distance from part (a), find the motorist's speed in ft/sec . Round to 1 decimal place. c. Determine the motorist's speed in mph. Round to the nearest mph.A large weather balloon is tethered by two ropes. One rope measures 23ft and attaches to the balloon at an angle of 32 from the ground. The second rope attaches to the base of the balloon at an angle of 15 with the ground. a. How far from the ground is the balloon floating? Round to the nearest tenth of a foot. b. Find the length of the second rope. Round to the nearest tenth of a foot c. If both ropes suddenly detach and the balloon rises straight up at a rate of 3ft/sec , how long will it take the balloon to reach a height of 50ft from the ground? Round to the nearest tenth of a second.The Duquesne Incline is a century-old railroad scaling Mt. Washington in Pittsburgh's south side neighborhood. The incline is 244m long and rises 120m in elevation. What is the angle of the incline to the nearest tenth of a degree? (See Example 4)The Jackson Hole Aerial Tram takes passengers from an elevation of 6311ft to an elevation of 10,450ft up the slope of Rendezvous Mountain in Wyoming. If the tram runs on a cable measuring 12,463ft in length, what is the angle of incline of the tram to the nearest tenth of a degree?To approximate the distance from the Earth to stars relatively close by. astronomers often use the method of parallax. Parallax is the apparent displacement of an object caused by a change in the observer's point of view. As the Earth orbits the Sun, a nearby star will appear to move against the more distant background stars. Astronomers measure a star's position at times exactly 6 months apart when the Earth is at opposite points in its orbit around the Sun. The Sun, Earth, and star form the vertices of a right triangle with PSE=90 . The length of is the distance between the Earth and Sun. approximately 92,900,000mi . The parallax angle (or simply parallax) is denoted by p . Use this information for Exercises 31-32. a. Find the distance between the Earth and Proxima Centauri (the closest star to the Earth beyond the Sun) if the parallax angle is 0.772 (arcseconds). Round to the nearest hundred billion miles, b. Write the distance in part (a) in light-years. Round to 1 decimal place. (Hint. 1 light-year is the distance that light travels in 1yr and is approximately 5.8781012mi .)To approximate the distance from the Earth to stars relatively close by. astronomers often use the method of parallax. Parallax is the apparent displacement of an object caused by a change in the observer's point of view. As the Earth orbits the Sun, a nearby star will appear to move against the more distant background stars. Astronomers measure a star's position at times exactly 6 months apart when the Earth is at opposite points in its orbit around the Sun. The Sun, Earth, and star form the vertices of a right triangle with PSE=90 . The length of is the distance between the Earth and Sun. approximately 92,900,000mi . The parallax angle (or simply parallax) is denoted by p . Use this information for Exercises 31-32. a. Find the distance between the Earth and Barnard's Star if the parallax angle is 0.547 arcseconds. Round to the nearest hundred billion miles. b. Write the distance in part (a) in light-years. Round to 1 decimal place. (Hint 1 light-year is the distance that light travels in 1yr and is approximately 5.8781012mi .)A sewer line must have a minimum slope of 0.25in. per horizontal foot but not more than 3in. per horizontal foot. A slope less than 0.25in. per foot will cause drain clogs, and a slope of more than 3in. per foot will allow water to drain without the solids. a. To the nearest tenth of a degree, find the angle of depression for the minimum slope of a sewer line. b. Find the angle of depression for the maximum slope of a sewer line. Round to the nearest tenth of a degree.In order for gravel roads to have proper drainage, the highest point on the road (the crown) should slope downward on either side to the shoulders of the road. EPA guidelines for maintaining gravel roads with a low volume of traffic suggest that a 20-ft wide road should have a centerline crown that is 5 to 7 in. high a. To the nearest tenth of a degree, find the angle of depression for a 5 -in. centerline crown. b. To the nearest tenth of a degree, find the angle of depression for a 7 -in. centerline crown.The weather satellite NOAA-15 orbits the Earth every 101min at an altitude of 807km as shown in the figure. The cameras on the satellite have a maximum range defined by the set of lines tangent to the Earth from the satellite. Using the radius of the Earth as 6357km , approximate the length of the arc from A to B to the nearest kilometer. This is the maximum distance along the surface of the Earth' seen" by the satellite's cameras.A communications satellite is in geosynchronous orbit. This means that its orbit coincides with the rotation period of the Earth so that it remains above a fixed point on the surface of the Earth at all times. The altitude of the satellite is 35,786km . The sender and receiver of a signal must both be within the line of sight of the satellite. What is the maximum distance along the surface of the Earth for which the sender and receiver can communicate? Take the radius of the Earth to be 6357km and round to the nearest kilometer.37PE38PE39PEFor Exercises 39-42, make a sketch to illustrate the bearing. S10WFor Exercises 39-42,, make a sketch to illustrate the bearing. S67EFor Exercises 39-42,, make a sketch to illustrate the bearing. N75EFor Exercises 43-46, find the bearing of the ship. Round to the nearest tenth of a degree if necessary.For Exercises 43-46, find the bearing of the ship. Round to the nearest tenth of a degree if necessary.For Exercises 43-46, find the bearing of the ship. Round to the nearest tenth of a degree if necessary.For Exercises 43-46, find the bearing of the ship. Round to the nearest tenth of a degree if necessary.For Exercises 47-50, find the bearing to the nearest tenth of a degree. A runner jogs 5mi east then 2mi south. What is the bearing from his starting point?For Exercises 47-50, find the bearing to the nearest tenth of a degree. A cyclist rides west for 7mi and then north 10mi . What is the bearing from her starting point?For Exercises 47-50, find the bearing to the nearest tenth of a degree. A boat leaves Matheson Hammock Marina at a constant speed of 3.5mph . The boat travels south for 36min and then east for 24min to a favorite fishing spot. After a day of fishing, find the bearing that the captain should use to travel back to the marina.For Exercises 47-50, find the bearing to the nearest tenth of a degree. A plane leaves Atlanta's Hartsfield Airport and flies north for 1hr and west for 2hr at an average speed of 400mph . Find the bearing that the plane should take for the return trip.For Exercises 51-54, round answers to the nearest unit. A fishing boat leaves a dock at a bearing of N34E . After running for 1.4hr at 25mph , how far north and east has it traveled? (See Example 6)52PEFor Exercises 51-54, round answers to the nearest unit. An airplane flying 310 knots has a bearing of S44W . After flying for 2.4hr , how many nautical miles (nmi) south and west has it traveled?For Exercises 51-54, round answers to the nearest unit. A ship traveling 6.4 knots has a bearing of N11W . After 3hr , how many nautical miles (nmi) north and west has it travelled?A ship leaves port at 8 knots heading N27W . After 2hr it makes a 90 clockwise turn to a new bearing of N63E and travels for 1.4hr . (See Example 7) a. Find the ship's distance from port to the nearest tenth of a nautical mile. b. Find the bearing required for the ship to return to port. Round to the nearest degree.56PEA boat leaves a dock at a bearing of N42W and travels 24mph for 1hr . The boat then makes a 90 left turn and travels 8mi to its destination. a. Find the exact distance from the dock to the boat's destination. b. Find the bearing required to locate the boat from the dock. Round to the nearest degree.58PEFor Exercises 59-60, find the lengths of the sides to the nearest tenth of a unit and the measures of the angles to the nearest tenth of a degree if necessary. A=C=AB=BC=AD=DB=For Exercises 59-60, find the lengths of the sides to the nearest tenth of a unit and the measures of the angles to the nearest tenth of a degree if necessary. ADC=ADB=BCD=BC=AC=AD=BD=AB=Find the measures of the sides and angles. Round angle measures to the nearest tenth of a degree. Give exact answers for the lengths of sides. ADC=CBA=DCB=BD=BC=A rectangular prism measures 6in. by 4in. by 3in . Find the angle formed by the diagonal of the prisim and the diagonal of the base, as shown. Round to 1 decimal place.Two children sit on either end of an 18-ft teeter totter. The center of the teeter totter is 2ft off the ground, and the position of each child is 8ft from the center. a. When one end of the teeter totter is on the ground, find the angle that the teeter totter makes with the ground. Round to the nearest tenth of a degree. b. When one end of the teeter totter is on the ground, how high is the child's seat at the other end? Round to 1 decimal placeA light fixture mounted 24ft above the ground illuminates a cone of light with an angle of 42 at the top. a. What is the radius of the circle of light on the ground? Round to 1 decimal place. b. How many square feet will be lit on the ground? Round to the nearest whole unit.For Exercises 65-66, refer to the table giving the angle of the Sun at noon at the first day of each season based on the latitude of the observer in the northern hemisphere. The term 23.5 represents the tilt of the Earth to the ecliptic plane (plane of the Earth's orbit around the Sun). In 1987, a group of protesters lead by Jacqueline kennedy Onassis defeated a plan to build two high rises that would have blocked sunlight to a significant swath of New York City’s Central Park. Eventually the design was scaled down and the buildings height reduced from appropriately 1020 ft to what is now the Time Warner Center at 750 ft. a. The latitude of New York City is 40.7N . Find the angle of elevation of the Sun at noon on the winter solstice (shortest day of the year). b. At noon on the winter solstice, how long is the shadow cast by a 1020-ft building? c. At noon on the winter solstice, how long is the shadow cast by a 750-ft building?For Exercises 65-66, refer to the table giving the angle of the Sun at noon at the first day of each season based on the latitude of the observer in the northern hemisphere. The term 23.5 represents the tilt of the Earth to the ecliptic plane (plane of the Earth's orbit around the Sun). The Earth is tilted 23.5 on its axis. Therefore, at noon on the summer solstice, the Earth is tilted 23.5 toward the Sun and all points on the Earth with latitude 23.5N will see the Sun directly overhead at noon. Points on the Earth with latitude 23.5N are on the Tropic of Cancer. Key West, Florida, has latitude 24.6 , just above the Tropic of Cancer. a. Find the angle of elevation of the Sun at noon in Key West on the first day of summer and on the first day of winter. b. Find the length of a shadow cast by a 105-ft building at noon on the first day of summer and on the first day of winter. Round to the nearest tenth of a foot. c. Find the length of a shadow cast by a 40-ft tree at noon on the first day of fall or spring.OSHA safety regulations require that the base of a ladder be placed 1ft from the wall for every 4ft of ladder length. To the nearest tenth of a degree, find the angle that the ladder forms with the ground and the angle that it forms with the wall.An A-frame storage shed is constructed such that the slanted roof line is always 3 times as long as the distance from the bottom corner of the roof to the centerline of the shed. To the nearest degree, what is the angle at the top of the roof?An airplane needs to fly to an airfield located 189 mi east and 568 mi north of its current location. To the nearest tenth of a degree, determine the bearing at which the plane should fly.Software used to program video games often uses an origin at the top left of the display canvas. The positive x-axis is to the right and the positive y-axis is downward. Suppose that a player moves on a direct path from the origin to a point P with pixel location 135,200 . Then the player moves directly to point Q at a pixel location of 420,150 . a. Find the player's bearing from the origin to point P . Round to one-hundredth of a degree. b. Find the player's bearing from point P to point Q . Round to one-hundredth of a degree.Two ships leave port at the same time. One travels N72E at 20 knots and the other travels S18E at a rate of 30 knots. a. After 2hr , how far apart are the two ships? Give an exact answer in nautical miles. b. Find the bearing from the ship farther to the north to the ship farther to the south. Round to the nearest tenth of a degree.72PEExplain how to represent the bearing from point P to point Q .What is meant by solving a triangle?A contractor building eight ocean-front condominiums wants to maximize the view of the ocean for each unit. The side of the building facing the ocean is not built in a straight line parallel to the ocean, but instead is built in a zigzag pattern as shown in the figure. Each condo has a window of length x facing the ocean at an angle of 70 from a line perpendicular to the ocean. a. Find the length x of each window. Round to the nearest foot. b. The windows facing the ocean are 8ft high and x feet wide. By using the zigzag pattern how much more ocean-front viewing area does each window provide than if the windows were parallel to the ocean? Round to the nearest square foot.A circle is inscribed within an isosceles triangle such that the circle is tangent to all three sides as shown in the figure. Using the fact that the radius r of the circle is perpendicular to any line tangent to the circle, find the value of r to the nearest tenth of an inch.A tool requires a V-shaped block to hold two rollers of radius 3cm and 2cm . Determine the angle of the "V" cut to the nearest tenth of a degree.Pipe for a water line must be installed from a main water line at point A to a building on Hontoon Island State Park at point B as shown in the figure. The cost to install water pipe over land is $10 per foot and the cost to install pipe under water is $20 per foot a. Write an expression in terms of 6 to represent the total cost c (in dollars) to lay pipe from point A to point B. b. Use the TABLE function on a calculator to find the cost for 6=20,25,30,35, and40 . Round to 1 decimal place. c. Which angle from part (b) yields the least cost? d. Using calculus, we can show that the angle needed to minimize the total cost is a solution to the equation 4000sectan2000sec2=0 . Solve the equation for6 , where 090.A woman participating in a triathlon can run 11ft/sec and swim 3ft/sec . She is at point A , 900ft from a straight shoreline and must swim to shore and run to point B , 3000ft down the beach. a. Write an expression representing the total time t (in seconds) for her to get from point A to point B as a function of . b. Use the TABLE function on a calculator to find the time t for =0,5,10,15,20, and 25. Round to 1 decimal place. c. Which angle from part (b) gives the shortest total time? d. Using calculus, we can show that the angle needed to minimize the total time is a solution to the equation 300sectan90011sec2=0 . Solve the equation for , where 090 Round to the nearest tenth of a degree.Solve ABC with B=70 , C=48, and b=15cm . Round the lengths of the sides to 1 decimal place.Solve ABC with A=33 , B=112 , and c=16 . Round the lengths of the sides to 1 decimal place.Solve ABC with A=49 , a=32 , and b=28 . Round the measures of the unknown angles and side to 1 decimal place.Solve ABC with B=31 , b=3 , a=6.2 . Round the measures of the unknown angles and side to 1 decimal place.Solve ABC with A=40 , a=22 , and b=25 . Round the measures of the unknown angles and the third side to 1 decimal place.A triangle has sides of 12m and 6m , and the angle between them is 76 . Find the area of the triangle. Round to 1 decimal place.To measures the maximum height of the water in a fountain, a trigonometry student measures the angle of elevation of the maximum point to be 25 . At a point 30ft farther away along the same line of sight, the angle of elevation is 18 . If the eye level of the student is 5.5ft , find the maximum height of the water. Round to the nearest tenth of a foot.An 80-ft flagpole is located at the top of a hill. An observer 220ft down the hill measures the angle formed by the bottom of the pole to the top of the pole to be 17.7 . Determine the angle of inclination of the hill to the nearest tenth of a degree.To apply the law of sines, an angle must be given and a side (opposite/adjacent) the given angle must be given, along with one other known side or angle.In which cases can the law of sines be used to solve a triangle? Choose from ASA,SAA,SSA,SSS,SAS .Suppose that ABC is a triangle with sides of length a,b , and c opposite angles A,B , and C , respectively. Write the law of sines.Suppose that the lengths of two sides a and b in a triangle are known along with the measure of the angle C between them. Then the area is given by Area=Suppose that we know the measure of angle A and the lengths of sides a and b of a triangle. If ab , how many possible triangles can be formed?Suppose that we know the measure of angle A and the lengths of sides a and b of a triangle. If bsinAab , how many possible triangles can be formed?For Exercises 7-12, solve the triangle. For the sides, give an expression for the exact value of the length and an approximation to 1 decimal place.For Exercises 7-12, solve the triangle. For the sides, give an expression for the exact value of the length and an approximation to 1 decimal place.For Exercises 7-12, solve the triangle. For the sides, give an expression for the exact value of the length and an approximation to 1 decimal place.For Exercises 7-12, solve the triangle. For the sides, give an expression for the exact value of the length and an approximation to 1 decimal place.For Exercises 7-12, solve the triangle. For the sides, give an expression for the exact value of the length and an approximation to 1 decimal place.For Exercises 7-12, solve the triangle. For the sides, give an expression for the exact value of the length and an approximation to 1 decimal place.For Exercises 13-20, solve ABC subject to the given conditions. Round the lengths of sides and measures of the angles to 1 decimal place if necessary. (See Examples 1-2) A=127,B=34,a=42For Exercises 13-20, solve ABC subject to the given conditions. Round the lengths of sides and measures of the angles to 1 decimal place if necessary. (See Examples 1-2) A=27,B=71,b=186For Exercises 13-20, solve ABC subject to the given conditions. Round the lengths of sides and measures of the angles to 1 decimal place if necessary. (See Examples 1-2) A=122.1,B=24.3,c=102.For Exercises 13-20, solve ABC subject to the given conditions. Round the lengths of sides and measures of the angles to 1 decimal place if necessary. (See Examples 1-2) A=10.6,C=98.8,b=59For Exercises 13-20, solve ABC subject to the given conditions. Round the lengths of sides and measures of the angles to 1 decimal place if necessary. (See Examples 1-2) A=34.7,C=68.1,c=43.918PE19PEFor Exercises 13-20, solve ABC subject to the given conditions. Round the lengths of sides and measures of the angles to 1 decimal place if necessary. (See Examples 1-2) A=53.8,B=60,c=179For Exercises 21-28, information is given about ABC . Determine if the information gives one triangle, two triangles, or no triangle. Solve the resulting triangle (s). Round the lengths of sides and measures of the angles to 1 decimal place if necessary. (See Examples 3-5) b=33,c=25,B=38For Exercises 21-28, information is given about ABC . Determine if the information gives one triangle, two triangles, or no triangle. Solve the resulting triangle (s). Round the lengths of sides and measures of the angles to 1 decimal place if necessary. (See Examples 3-5) b=5,c=12,C=73For Exercises 21-28, information is given about ABC . Determine if the information gives one triangle, two triangles, or no triangle. Solve the resulting triangle (s). Round the lengths of sides and measures of the angles to 1 decimal place if necessary. (See Examples 3-5) a=185,c=132,C=63For Exercises 21-28, information is given about ABC . Determine if the information gives one triangle, two triangles, or no triangle. Solve the resulting triangle (s). Round the lengths of sides and measures of the angles to 1 decimal place if necessary. (See Examples 3-5) b=6,c=12,B=38For Exercises 21-28, information is given about ABC . Determine if the information gives one triangle, two triangles, or no triangle. Solve the resulting triangle (s). Round the lengths of sides and measures of the angles to 1 decimal place if necessary. (See Examples 3-5) a=13,b=18,A=45For Exercises 21-28, information is given about ABC . Determine if the information gives one triangle, two triangles, or no triangle. Solve the resulting triangle (s). Round the lengths of sides and measures of the angles to 1 decimal place if necessary. (See Examples 3-5) a=3,b=1,B=17For Exercises 21-28, information is given about ABC . Determine if the information gives one triangle, two triangles, or no triangle. Solve the resulting triangle (s). Round the lengths of sides and measures of the angles to 1 decimal place if necessary. (See Examples 3-5) a=132.5,b=108.2,B=13.1For Exercises 21-28, information is given about ABC . Determine if the information gives one triangle, two triangles, or no triangle. Solve the resulting triangle (s). Round the lengths of sides and measures of the angles to 1 decimal place if necessary. (See Examples 3-5) a=48.8,c=39.9,C=22For Exercises 29-34, find the area of a triangle with the given measurements. Round to 1 decimal place. (See Example 6) A=107,b=17ft,c=3ftFor Exercises 29-34, find the area of a triangle with the given measurements. Round to 1 decimal place. (See Example 6) A=143,b=20m,c=16mFor Exercises 29-34, find the area of a triangle with the given measurements. Round to 1 decimal place. (See Example 6) B=98.8,a=2.1in.,c=5.3in.For Exercises 29-34, find the area of a triangle with the given measurements. Round to 1 decimal place. (See Example 6) B=2.6,a=3.5cm,c=10.8cmFor Exercises 29-34, find the area of a triangle with the given measurements. Round to 1 decimal place. (See Example 6) C=74.6,a=13mm,b=23mmFor Exercises 29-34, find the area of a triangle with the given measurements. Round to 1 decimal place. (See Example 6) C=125.7,a=178mi,b=239miThe area in the Atlantic Ocean known as the Bermuda Triangle is defined by an imaginary triangle connecting Miami, Florida; San Juan, Puerto Rico; and the island of Bermuda. Measuring on a map, the distance from both Miami to San Juan and from Miami to Bermuda is approximately 1033mi . Assuming that the angle from Bermuda to Miami to San Juan is approximately 65 , what is the area of the Bermuda Triangle? Round to the nearest square mile.The triangular face of a gabled roof measures 33.8ft on each sloping side with an angle of 134.8 at the top of the roof. What is the area of the face? Round to the nearest square foot.Two fire lookout towers at points A and B are 2mi apart. Find the distances from points A and B to a fire at point C if ABC is 7848 and BAC is 8436 . Round to the nearest tenth of a mile.A helicopter is on a path directly overhead line AB when it is simultaneously observed from locations A and B separated by 900ft . The angle of elevation from A is 4230 and the angle of elevation from B is 3012 . a. What is the distance from each location to the helicopter? Round to the nearest foot. b. How high is the helicopter from the ground at the moment of observation? Round to the nearest foot.A surveyor wants to measure the distance across a lake from a point A due north to a point B . He runs a 1500-ft line from point A to point C in the direction N52E . He then runs a second line from C to B in the direction N4124W . a. To the nearest foot, how long is the second line? b. To the nearest foot what is the distance between A and B ?40PEFrom a point along a straight road, the angle of elevation to the top of a hill is 33 . From 300ft farther down the road, the angle of elevation to the top of the hill is 24 . How high is the hill? Round to the nearest foot. (See Example 7)An observer on the ground sites a plane at an angle of elevation of 41.2 . At the same time, a second observer 3000m farther away along the same line of site measures the angle of elevation as 35 . How high is the plane? Round to the nearest 100m .A wire is fastened to a point T on a tree and to point A located 8.8ft from the base of the tree along level ground (see figure). The angle that the wire makes with level ground is 44 , and the tree leans 12 from vertical away from point A . How high off the ground is the point where the wire is fastened to the tree? Round to the nearest tenth of a foot.The Leaning Tower of Suurhusen is a medieval steeple in Suurhasen, Germany. The tower leans at an angle of 5.1939 from the vertical. (In comparison, the Leaning Tower of Pisa leans at an angle of 3.97 .) The angle of elevation to the top of the tower is 44.2 when measured 100ft from the base of the tower. Find the distance h from the base of the tower to the top of the tower. Round to the nearest tenth of a foot.45PEA mountain cabin is built on the side of a hill with a porch extending over a portion of the downhill slope. An observer 20ft downhill from one of the 8-ft vertical support beams measures the angle formed from the top of the beam to the bottom as 10 . Find the angle of inclination of the hill to the nearest tenth of a degree.Two residential buildings are to be constructed with a grassy recreational area between them. The taller building is 700ft high. From the roof of the shorter building, the angle of elevation to the top of the taller building is 78 and the angle of depression to the base of the taller building is 62 . a. How tall is the shorter building? Round to the nearest foot b. What is the distance between the buildings? Round to the nearest footA hiker wants to estimate the height of a mountain before attempting a climb to the top. Her first measurement shows an angle of elevation to the top of the mountain as 63.4 . Her second measurement, taken 950ft closer to the base of the mountain, yields an angle of elevation of 75.3 . From these measurements, estimate the height of the mountain. Round to the nearest hundred feet.A surveyor fixes a 420-ft baseline between points A and B , where B is due east of A . The bearing from A to a third point P is N52E and the bearing from B to P isN27W . Find the perimeter and area of the triangular plot of land formed by A,B , and P . Round the perimeter to the nearest foot and the area to the nearest hundred square feet.A surveyor fixes a 200-ft baseline between points A and B , where B is due east of A The bearing from A to a third point P is S37W and the bearing from B to P isS62W . Find the perimeter and area of the triangular plot of land formed by A,B , and P . Round the perimeter to the nearest foot and the area to the nearest 100 square feet.The connector rod from the piston to the crankshaft in a certain 2.0-L engine is 6.4in . The radius of the crank circle is 2.8in . If the angle made by the connector rod with the horizontal at the wrist pin P is 20, how far is the wrist pin from the center C of the crankshaft? Round to the nearest tenth of an inch.Two planets follow a circular orbit around a central star in the same plane. The distance between the star at point S and one planet at point A is 135 million miles. The distance between the star and the other planet at point B is 100 million miles. If an observer on the first planet at point A sights the second planet such that SAB=42 find the distance between the planets. Round to the nearest million miles.For Exercises 53-66, solve ABC . Round the lengths of sides and measures of the angles to 1 decimal place if necessary. A=117,B=32,b=8.2For Exercises 53-66, solve ABC . Round the lengths of sides and measures of the angles to 1 decimal place if necessary. a=42,c=60,C=17For Exercises 53-66, solve ABC . Round the lengths of sides and measures of the angles to 1 decimal place if necessary. b=6,c=8,B=24For Exercises 53-66, solve ABC . Round the lengths of sides and measures of the angles to 1 decimal place if necessary. A=103.1,B=13,c=3.8For Exercises 53-66, solve ABC . Round the lengths of sides and measures of the angles to 1 decimal place if necessary. B=13.3,C=68.4,c=12For Exercises 53-66, solve ABC . Round the lengths of sides and measures of the angles to 1 decimal place if necessary. a=11,c=3,C=142For Exercises 53-66, solve ABC . Round the lengths of sides and measures of the angles to 1 decimal place if necessary. a=118,c=112,C=24.6For Exercises 53-66, solve ABC . Round the lengths of sides and measures of the angles to 1 decimal place if necessary. B=47.8,C=99.3,a=183For Exercises 53-66, solve ABC . Round the lengths of sides and measures of the angles to 1 decimal place if necessary. a=16,b=15.1,A=113For Exercises 53-66, solve ABC . Round the lengths of sides and measures of the angles to 1 decimal place if necessary. b=15.1,c=18.6,B=113For Exercises 53-66, solve ABC . Round the lengths of sides and measures of the angles to 1 decimal place if necessary. A=4.6,C=1.2,a=23For Exercises 53-66, solve ABC . Round the lengths of sides and measures of the angles to 1 decimal place if necessary. B=12,C=136,b=800For Exercises 53-66, solve ABC . Round the lengths of sides and measures of the angles to 1 decimal place if necessary. a=325,c=221,C=78.8For Exercises 53-66, solve ABC . Round the lengths of sides and measures of the angles to 1 decimal place if necessary. A=153,C=2,b=2After a hurricane, a homeowner examines trees on his property for damage. He thinks a 20-ft palm tree is leaning slightly from its original vertical position. From a point 23ft away, he measures the angle of elevation to the top of the palm tree as 43 . Is the palm tree leaning? If so, by how many degrees from the vertical? Round to the nearest tenth of a degree.A company manufactures pennants in the shape of an isosceles triangle. The long sides of each triangle are 18in , and the angle between the long sides is 40. The weatherproof fabric from which the pennants are made costs6.95/yd2 . How much will it cost the company to make 10,000 pennants? Round to the nearest dollar.Consider a sector of a circle of radius 6cm with central angle 112. a. Find the area of the triangular region shaded in green. Round to the nearest hundredth of a square centimeter. b. Find the area of the region in yellow bounded by the circle and the chord of the circle.Refer to the figure. a. Write expressions for the exact values of a,b , and c . b. Approximate the values of a,b , and c to the nearest tenth of a centimeter.Refer to the figure. a. Write expressions for the exact values of a,b , and c . b. Approximate the values of a,b , and c to the nearest tenth of an inch.Given ABC with sides of lengths a,b , and c opposite angles A,B , and C , respectively, use the law of sines to show that a+bbsinA+sinBsinB73PEGiven SSA (the ambiguous case), outline the four possible scenarios regarding the number of triangles that are possible.List various methods to confirm your answers after solving a triangle using the law of sines.A triangle ABC has sides a,b , and c across from angles A,B , and C , respectively. Explain the relationship given by Area=12bcsinA .From the figure, show that sin45=23sin95 .Use the law of sines in the form sinAsinB=ab 1 to prove the law of tangents: aba+b=tanAB2tanA+B2 .Use the relationships ac=sinAsinC and bc=sinBsinA from the law the for law this of sines to show that a+bc=cosAB2sinC2 . This is called a Mollweide equation, named after German mathematician and astronomer Karl Mollweide 1774-1825 . Notice that the equation uses all six parts of a triangle, and for this reason, is sometimes used to check the solution to a triangle.Given ABC circumscribed by a circle, show that the diameter d of the circle is d=bsinB radii from the center of the circle to points A and C. Use a theorem from geometry relating the measure of an inscribed angle of a circle to the corresponding central angle.) Note that it follows from the law of sines that d=asinA=bsinB=csinCSolve ABC with B=61 , c=19 , and a=28 . Round the measures of the unknown side and angles to 1 decimal place.Solve ABC with a=62 , b=48 , and c=41 . Round the measures of the angles to the nearest tenth of a degree.Repeat Example 3 with the initial bearing of the Coast Guard ship N17W at a speed of 18mph for 3hr .Find the area of a triangle with sides of length a=15.4m , b=22.6m , and c=26m . Round to the nearest square meter.In which situation (s) can the law of cosines be used to solve for the remaining angles and sides of an oblique triangle? Choose from AAS,SAS,ASS,SSS .Given ABC with sides a,b , and c opposite vertices A,B , and C , write an expression for a2 in terms of the lengths of the other sides and opposite angle.Given ABC with sides a,b , and c opposite vertices A,B , and C , write an expression for cos B in terms of the lengths of the sides of the triangle.Given a triangle of sides of lengths a,b , and c , Heron's formula gives the area of the triangle as where s represents half the perimeter of the triangle. That is s= .For Exercises 5-22, solve ABC subject to the given conditions if possible. Round the lengths of sides and measures of the angles (in degrees) to 1 decimal place if necessary. (See Example 1-2)For Exercises 5-22, solve ABC subject to the given conditions if possible. Round the lengths of sides and measures of the angles (in degrees) to 1 decimal place if necessary. (See Example 1-2)For Exercises 5-22, solve ABC subject to the given conditions if possible. Round the lengths of sides and measures of the angles (in degrees) to 1 decimal place if necessary. (See Example 1-2)For Exercises 5-22, solve ABC subject to the given conditions if possible. Round the lengths of sides and measures of the angles (in degrees) to 1 decimal place if necessary. (See Example 1-2)For Exercises 5-22, solve ABC subject to the given conditions if possible. Round the lengths of sides and measures of the angles (in degrees) to 1 decimal place if necessary. (See Example 1-2)For Exercises 5-22, solve ABC subject to the given conditions if possible. Round the lengths of sides and measures of the angles (in degrees) to 1 decimal place if necessary. (See Example 1-2)For Exercises 5-22, solve ABC subject to the given conditions if possible. Round the lengths of sides and measures of the angles (in degrees) to 1 decimal place if necessary. (See Example 1-2) a=28.3,c=17.4,B=11.3For Exercises 5-22, solve ABC subject to the given conditions if possible. Round the lengths of sides and measures of the angles (in degrees) to 1 decimal place if necessary. (See Example 1-2) b=89.2,c=23.1,A=108For Exercises 5-22, solve ABC subject to the given conditions if possible. Round the lengths of sides and measures of the angles (in degrees) to 1 decimal place if necessary. (See Example 1-2) a=15,b=12,c=15For Exercises 5-22, solve ABC subject to the given conditions if possible. Round the lengths of sides and measures of the angles (in degrees) to 1 decimal place if necessary. (See Example 1-2) a=25,b=30,c=35For Exercises 5-22, solve ABC subject to the given conditions if possible. Round the lengths of sides and measures of the angles (in degrees) to 1 decimal place if necessary. (See Example 1-2) a=27,c=26,B=67.8For Exercises 5-22, solve ABC subject to the given conditions if possible. Round the lengths of sides and measures of the angles (in degrees) to 1 decimal place if necessary. (See Example 1-2) a=40.5,b=38.1,C=73.217PEFor Exercises 5-22, solve ABC subject to the given conditions if possible. Round the lengths of sides and measures of the angles (in degrees) to 1 decimal place if necessary. (See Example 1-2) a=500,b=200,c=400For Exercises 5-22, solve ABC subject to the given conditions if possible. Round the lengths of sides and measures of the angles (in degrees) to 1 decimal place if necessary. (See Example 1-2) b=146.8,c=122.7,A=110.4For Exercises 5-22, solve ABC subject to the given conditions if possible. Round the lengths of sides and measures of the angles (in degrees) to 1 decimal place if necessary. (See Example 1-2) b=802.5,c=436.1,A=103.721PEFor Exercises 5-22, solve ABC subject to the given conditions if possible. Round the lengths of sides and measures of the angles (in degrees) to 1 decimal place if necessary. (See Example 1-2) a=18,b=32,c=10A boat leaves port and follows a course of N77E at 9 knots for 3hr and 20min . Then, the boat changes to a new course of S28E at 12 knots for 5hr . a. How far is the boat from port? b. Suppose that the boat becomes disabled. How long will it take a rescue boat to arrive if the rescue boat leaves from port and travels 18 knots? Round to the nearest minute. c. What bearing should the rescue boat follow?A fishing boat leaves a marina and follows a course of S62W at 6 knots for 20min . Then the boat changes to a new course of S30W at 4 knots for 1.5hr . a. How far is the boat from the marina? b. What course should the boat follow for its return trip to the marina?Two planes leave the same airport. The first plane leaves at 1:00P.M . and averages 480mph at a bearing ofS62E . The second plane leaves at 1:15P.M and averages 410mph at a bearing ofN12W . a. How far apart are the planes at 2:45P.M. ? b. What is the bearing from the first plane to the second plane at that time? Round to the nearest degree.Two boats leave a marina at the same time. The first boat travels 6 knots at a bearing of N39E , and the second boat travels 4 knots at a bearing of S87W . a. How far apart are the boats at the end of 2hr ? b. What is the bearing from the first boat to the second boat at that time?A 40-ft boom on a crane is attached to the crane platform at point D . A cable is attached to the end of the boom at point B and to a 12-ft A-frame at point A anchored by the counter weight. If BDA is 85.2 , find the length of the cable to the nearest tenth of a foot.28PEA regulation fast-pitch softball diamond for high school competition is a square, 60ft on a side. The pitcher's mound is colinear with home plate and second base. Furthermore, the distance from the back of home plate to the center of the pitcher's mound is 43ft to the nearest tenth of a foot, find the distance between a. The pitcher's mound and first base. b. The pitcher's mound and second base.The Flatiron Building in New York City is said to have gotten its name because its cross-section resembles the shape of the household appliance of the same name. The perimeter of the building is bounded by Broadway, 5th Avenue, and East 22nd and 23rd Streets. Using side lengths of approximately 87ft,190ft , and 173ft , determine the angle at the "point" of the Flatiron Building at the comer of 5th Avenue and East 23rd Street. Round to the nearest tenth of a degree.The distance between Dallas, Texas, and Atlanta, Georgia, on a map is 11.75in . The distance from Atlanta to Chicago, Illinois, is 9.5in. , and the distance from Chicago to Dallas is 13.25in. If the bearing from Dallas to Atlanta is N85E , find the bearing from Chicago to Dallas. Round to the nearest tenth of a degree.A bicyclist is at point A on a paved road and must ride to point C on another paved road. The two roads meet at an angle of 38 at point B The distance from A to B is 18mi , and the distance from B to C is 12mi (see the figure), if the bicyclist can ride 22mph on the paved roads and 6.8mph off-road, would it be faster for the bicyclist to ride from A to C on the paved roads or to ride a direct line from A to C off-road? Explain.A 150-ft tower is anchored on a hill by two guy wires. The angle of elevation of the hill is 8 . Each guy wire extends from the top of the tower to a ground anchor 60ft from the base of the tower and in line with the tower. Find the length of each guy wire. Round to the nearest foot.A 75-ft tower is located on the side of a hill that is inclined 26 to the horizontal. A cable is attached to the top of the tower and anchored uphill a distance of 35ft from the base of the tower. Find the length of the cable. Round to the nearest foot.In Exercises 35-38, use Heron's formula to find the area of the triangle with sides of the given lengths. Round to the nearest tenth of a square unit. (See Example 4) a=13in.,b=7in.,c=8inIn Exercises 35-38, use Heron's formula to find the area of the triangle with sides of the given lengths. Round to the nearest tenth of a square unit. (See Example 4) a=10mi,b=17mi,c=9miIn Exercises 35-38, use Heron's formula to find the area of the triangle with sides of the given lengths. Round to the nearest tenth of a square unit. (See Example 4) a=18.6cm,b=12.3cm,c=25.9cmIn Exercises 35-38, use Heron's formula to find the area of the triangle with sides of the given lengths. Round to the nearest tenth of a square unit. (See Example 4) a=7.9yd,b=12.1yd,c=19,3ydA triangular sail for a Bolger Tortise sailboat (named after boat designer Phil Bolger) has sides of length 81,101, and 910 What is the sail area?Find the area of the grassy region from Exercise 28. Round to the nearest square foot.A farmer wants to fertilize a triangular field with sides of length 300yd,220yd , and 180yd . Fertilizer costs 180 per acre 1acre=4840yd2 . Furthermore, the time required to fertilizer 1 acre is approximately 2.8hr with combined labor and equipment costs of $23.00 per hour. a. To the nearest acre, how big is the field? b. Using the number of acres from part (a), estimate the cost to fertilize the field. Round to the nearest dollar.An artist lays down a white triangular background on a building on which she will paint a Penrose triangle. The triangular area measures 20ft on each side and will be covered with three layers of primer paint. a. To the nearest square foot, what is the area of the triangular region to be primed? b. If primer paint covers 200ft2/gal , how many gallons of primer must be purchased?For Exercises 43-46, the vertices of a triangle are defined by the given points. To the nearest tenth, determine a. the perimeter of the triangle. b. the area of the triangle. c. the measure of the angles in the triangle. A2,2,B4,7,C8,1For Exercises 43-46, the vertices of a triangle are defined by the given points. To the nearest tenth, determine a. the perimeter of the triangle. b. the area of the triangle. c. the measure of the angles in the triangle. A1,5,B5,8,C10,3For Exercises 43-46, the vertices of a triangle are defined by the given points. To the nearest tenth, determine a. the perimeter of the triangle. b. the area of the triangle. c. the measure of the angles in the triangle. A3,2,B1,5,C6,1For Exercises 43-46, the vertices of a triangle are defined by the given points. To the nearest tenth, determine a. the perimeter of the triangle. b. the area of the triangle. c. the measure of the angles in the triangle. A2,1,B10,2,C5,4Three mutually tangential circles with radii located at A,B , and C , have radii of2ft,3ft , and4ft , respectively. a. Find the area of ABC . Round to the nearest tenth of a square foot. b. Find the measures of angles A,B , and C . Round to the nearest tenth of a degree. c. Find the area of the shaded region. Round to the nearest tenth of a square foot48PEA parallelogram has adjacent sides of 12.2cm and 9.6cm and the included angle is 62.5 . To the nearest tenth of a centimeter, a. Find the length of the shorter diagonal. b. Find the length of the longer diagonal.A regular octagon (8- sided figure with sides of equal length) is inscribed in a circle of radius 10in. Find the perimeter of the octagon to the nearest inch.Why is the law of sines not an option to solve a triangle given SAS or SSS?To solve a triangle given SAS , the law of cosines is used first to find the length of the side opposite the known angle. Then, if you choose to use the law of sines to solve for one of the two remaining angles, the guidelines suggest solving for the angle opposite the shorter of the two given sides. Why is this the case?Given the lengths of the three sides of a triangle SSS , why is the law of cosines used to find the measure of the largest angle first?