Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Calculus and Its Applications (11th Edition)
Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the x-value at which each extremum occurs. When no interval is specified, use the real numbers, (,). f(x)=2x3;[1,5)Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the x-value at which each extremum occurs. When no interval is specified, use the real numbers, .
78.
Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the x-value at which each extremum occurs. When no interval is specified, use the real numbers, .
79.
80E81E82E83E84E85E86E87E88E89E90E91E92ECheck Exercises 49, 51, 53, 57, 61, 65, 67, 69, 73, and 85 with a graphing calculator. f(x)=x2+432x;(0,)Check Exercises 49, 51, 53, 57, 61, 65, 67, 69, 73, and 85 with a graphing calculator. f(x)=2x4+x;[1,1]Check Exercises 49, 51, 53, 57, 61, 65, 67, 69, 73, and 85 with a graphing calculator. f(x)=(x1)3Check Exercises 49, 51, 53, 57, 61, 65, 67, 69, 73, and 85 with a graphing calculator. t(x)=x42x2Monthly productivity. An employees monthly productivity M, in number of units produced, is found to be a function of t, the number of years of employment. For a certain product, a productivity function is given by M(t)=2t2+100t+180,0t40. Find the maximum productivity and the year in which it is achieved.98. Advertising. Sound Software estimates that it will sell N units of a program after spending a dollars on advertising, where
and a is in thousands of dollars. Find the maximum number of units that can be sold and the amount that must be spent on advertising in order to achieve that maximum.
99E100. Labor. The percentage of women aged 21–54 in the U.S. civilian labor force can be modeled by
where x is the number of years since 1992. (Source: based on data from www.bls.gov.) According to this model, in what year during the period 1992–2012 was this percentage a maximum?
101. Worldwide oil production. One model of worldwide oil production is
where is the number of barrels, in thousands, produced t years after 2000. (Source: Based on data from the U.S. Energy Information Administration.) According to this model, in what year did worldwide oil production achieve an absolute minimum? What was that minimum?
Maximizing profit. Corner Stone Electronics determines that its weekly profit, in dollars, from the production and sale of x amplifiers is P(x)=1500x26x+10. Find the number of amplifiers, x, for which the total weekly profit is a maximum. Maximum: $1500 at x=3 amplifiers Maximizing Profit. The total-cost and total-revenue functions for producing x items are C(x)=5000+600xandR(x)=12x2+1000x, where 0x600. Use these functions for Exercises 103 and 104.a. Find the total-profit function P(x). P(x)=12x2+400x5000 b. Find the number of items, x, for which total profit is a maximum.104. a. Average profit is given by. Find.
b. Find the number of items, x, for which average profit is a maximum.
105. Blood pressure. For a dosage of x cubic centimeters (cc) of a certain drug, the resulting blood pressure B is approximated by
.
Find the maximum blood pressure and the dosage at which it occurs.
106. How is the second derivative useful in finding the absolute extrema of a function?
107EFor Exercises 107–110, find the absolute maximum and minimum values of each function, and sketch the graph.
108.
109EFor Exercises 107–110, find the absolute maximum and minimum values of each function, and sketch the graph.
110.
111. Consider the piecewise-defined function defined by;
a. Sketch its graph.
b. Identify the absolute maximum.
c. What is the absolute minimum?
112E113EFind the absolute maximum and minimum values of the function, if they exist, over the indicated interval. h(x)=x1x;[0,1]115. Business: total cost. Certain costs in business can be separated into two components: those that increase with volume and those that decrease with volume. For example, customer service becomes more expensive as its quality increases, but part of the increased cost is offset by fewer customer complaints.
Katie’s Clocks determines that its cost of service, , in thousands of dollars, is modeled by
where x represents the number of “quality units.” Find the number of “quality units” that the firm should use in order to minimize its total cost of service.
116. Let
.
For what value of x is y a minimum?
117. Business: worldwide oil production. Refer to Exercise 101. In what year(s) during the period 2000–2008 was worldwide oil production increasing most rapidly, and at what rate was it increasing?
118. How is the first derivative useful in finding the absolute extrema of a function?
119. Business: U.S. oil production. One model of oil production in the United States is given by
where is the number of barrels of oil, in billions, produced in a year, t years after 1910. (Source: Beyond Oil, by Kenneth S. Deffeyes, p. 41, Hill and Wang, New York, 2005.)
a. According to this model, what is the absolute maximum amount of oil produced in the United States and in what year did that production occur?
b. According to this model, at what rate was United States oil production declining in 2010 and in 2015?
Graph each function over the given interval. Visually estimate where any absolute extrema occur. Then use the TABLE feature to refine each estimate.
120.
Graph each function over the given interval. Visually estimate where any absolute extrema occur. Then use the TABLE feature to refine each estimate. f(x)=34(x21)2/3;[12,)Graph each function over the given interval. Visually estimate where any absolute extrema occur. Then use the TABLE feature to refine each estimate.
122.
1. Of all numbers whose sum is 70, find the two that have the maximum product That is, maximize, .
2. Of all numbers whose sum is 50, find the two that have the maximum product That is, maximize, .
3. Of all numbers whose difference is 6. Find the two that have the minimum product.
Of all numbers whose difference is 4, find the two that have the minimum product.Maximize Q=xy2 where x and y are positive numbers such that x+y2=4Maximize Q=xy2 where x and y are positive numbers such that x+y2=1MinimizeQ=x2+2y2,wherex+y=38.
Maximum Q=xy, where x and y are positive numbers such that x+43y2=1Maximize Q=xy. Where x and y are positive numbers such that 43x2+y=1611. Maximizing area.. A rancher wants to enclose two rectangular areas near a river, one for sheep and one for cattle. There are 240 yd of fencing available. What is the largest total area that can be enclosed?
Maximizing area. A lifeguard needs to rope off a rectangular swimming area in front of Long Lake Beach, using 180 yd of rope and floats. What dimensions of the rectangle will maximize the area? What is the maximum area? (Note that the shoreline is one side of the rectangle.)13. Maximizing area. Hentz Industries plans to enclose three parallel rectangular areas for sorting returned goods The three areas are within one large rectangular area and 1200 yd of fencing is available What is the largest total area that can be enclosed?
14. Maximizing area. Grayson Farms plans to enclose three parallel rectangular livestock pens within one large rectangular area using 600 ft of fencing One side of the enclosure is a pre-existing stone wall.
a. If the three rectangular pens have their longer sides parallel to the stone wall, find the largest possible total area that can be enclosed
b. If the three rectangular pens have their shorter sides perpendicular to the stone wall, find the largest possible total area that can be enclosed
15. Maximizing area. Of all rectangles that have a perimeter of 42 ft, find the dimensions of the one with the largest area. What is its area?
16. Maximizing area. A carpenter is building a rectangular shed with a fixed perimeter of 54 ft. What are the dimensions of the largest shed that can be built? What is its area?
Maximizing volume. From a thin piece of cardboard 20 in, by 20 in., square comers are cut out so that the sides can be folded up to make a box What dimensions will yield a box of maximum volume? What is the maximum volume?Maximizing volume. From a 50-cm-by-50-cm sheet to aluminum, square corners are cut out so that the sides can be folded up to make a box. What dimensions will yield a box of maximum volume? What is the maximum volume?19. Minimizing surface area. Mendoza Soup Company is constructing an open-top, square-based, rectangular metal tank that will have a volume of what dimensions will minimize surface area? What is the minimum surface area?
Minimizing surface area. Drum Tight Containers is designing an open-top square-based rectangular box that will have a volume of 62.5in3. What dimensions will minimize surface area? What is the minimum surface area?Minimizing surface area. Open Air Waste Management is designing a rectangular construction dumpster that will be twice as long as it is wide and must hold 12yd3 of debris. Find the dimensions of the dumpster that will minimize its surface areaMinimizing surface area. Ever Green Gardening is designing a rectangular compost container that will be twice as tall as it is wide and must hold 18ft3 of composted food scraps. Find the dimensions of the compost container with minimal surface area (include the bottom and top)For Exercises 23-28 find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit, Assume that revenue, Exercises 23-26
23.
For Exercises 23-28 find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit, Assume that revenue, R(x),andcost,C(x),areindollarsfor Exercises 23-26 R(x)=50x0.5x2,C(x)=10x+3For Exercises 23-28 find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit, Assume that revenue, R(x),andcost,C(x),areindollarsfor Exercises 23-26 R(x)=2x,C(x)=0.01x2+0.6x+30For Exercises 23-28 find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit, Assume that revenue, R(x),andcost,C(x),areindollarsfor Exercises 23-26 R(x)=5x,C(x)=0.001x2+1.2x+60For Exercises 23-28 find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit, Assume that revenue, Exercises 23-26
27. ; assume that are in thousands of dollars, and x is in thousands of units.
For Exercises 23-28 find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit, Assume that revenue, Exercises 23-26
28. are in thousands of dollars, and x is in thousands of units
Maximizing profit. Riverside Appliances is marketing a new refrigerator. It determines that in order to sell x refrigerators, the price per refrigerator must be p=2800.4x. It also determines that the total cost of producing x refrigerators is given by C(x)=5000+0.6x2. a. Find the total revenue, R(x) b. Find the total profit, P(x) c. How many refrigerators must the company produce and sell in order to maximize profit? d. What is the maximum profit? e. What price per refrigerator must be charged in order to maximize profit?Maximizing profit. Raggs, Ltd., a clothing firm, determines that in order to sell x suits, the price per suit must be P=1500.5x. It also determines that the total cost of producing x suits is given by C(x)=4000+0.25x2. a. Find the total revenue, R(x) b. Find the total profit, P(x) c. How many suits must the company produce and sell in order to maximize profit? d. What is the maximum profit? e. What price per suit must be charged in order to maximize profit?Maximizing profit. Gritz-Charlston is a 300-unit luxury hotel All rooms are occupied when the hotel. Charges $80 per day for a room. For every increase of x dollars in the daily room rate, there are x rooms vacant Each occupied room costs $22 per day to service and maintain What should the hotel charge per day in order to maximize profit?32. Maximizing revenue. Edwards University wants to determine what price to charge for tickets to football games, At a price of $18 per ticket, attendance averages 40,000 people per game Every decrease of $3 to the ticket price adds 10,000 people to the average attendance Every person at a game spends an average of $4.50 on concessions What price per ticket should be charged to maximize revenue? How many people will attend at that price?
Maximizing parking tickets. Oak Glen currently employs 8 patrol officers who each write an average of 24 parking tickets per day For every additional officer placed on patrol, the average number of parking tickets per day written by each officer decreases by 4 How many additional officers should be placed on patrol in order to maximize the number of parking tickets written per day?Maximizing yield. Hood Apple Farm yields an average of 30 bushels of apples per tree when 20 trees are planted on an acre of ground. If 1 more tree is planted per acre, the yield decreases by 1 bushel (bu) per tree as a result of crowding. How many trees should be planted on an acre in order to get the highest yield?35. Nitrogen prices. During 2001, nitrogen prices fell by 41%. Over the same year, nitrogen demand went up by 12%. (Source: Chemical week.)
a. Assuming a linear change in demand, find the demand function, , by finding the equation of the line that passes through the points . Here x is the price as a fraction of the January 2001 price, and is the demand as a fraction of the demand in January.
b. As a percentage of the January 2001 price, what should the price of nitrogen be to maximize revenue?
36. Vanity license plates. According to a pricing model, increasing the fee for vanity license plates by $1 decreases the percentage of a state’s population that will request them by 0.04% (Source: E. D. Craft, “The demand for vanity (plates): Elasticities, net revenue maximization, and deadweight loss, “ Contemporary Economic Policy, )
a. Recently, the tee for vanity license plates in Maryland was $25, and the percentage of the state’s population that had vanity plates was 2 13% Use this information to construct the demand function, , for the percentage of Maryland’s population that will request vanity license plates for a fee of x dollars.
b. Find the fee, x, that will maximize revenue from vanity plates
Maximizing revenue. When the Marchant Theater charges $5 for admission, there is an average attendance of 180 people For every $0, 10 increase in admission, there is a loss of 1 customer from the average number What admission should be charged in order to maximize revenue?Minimizing costs. A rectangular box with a volume of 320ft3 is to be constructed with a square base and top The cost per square foot for the bottom is 15,forthetopis10,andforthesidesis2.5. What dimensions will minimize the cost?39. Minimizing cost. A rectangular parking area measuring is to be enclosed on three sides using chain-link fencing that costs $4.50 per foot The fourth side will be a wooden fence that costs $7 per foot What dimensions will minimize the total cost to enclose this area, and what is the minimum cost (rounded to the nearest dollar)?
Minimizing cost. A rectangular garden measuring 1200yd2 is to be enclosed on two parallel sides by stone wall that costs $35 per yd and the other two sides by wooden fencing that costs $28 per yd What dimensions will minimize the total cost of enclosing this garden, and what is the minimum cost (rounded to the nearest dollar)?41. Maximizing area. Bradley Publishing decides that each page in a new book must have an area of margin at the top and at the bottom of each page, and a 0.5-in margin on each of the sides. What should the outside dimensions of each page be so that the printed area is a maximum?
Minimizing inventory costs. A sporting goods store sells 100 pool tables per year It costs $20 to store one pool table for a year, To reorder, there is a fixed cost of $40 per shipment plus $16 for each pool table How many times per year should the store order pool tables, and in what lot size, in order to minimize inventory costs?43. Minimizing inventory costs. A pro shop in a bowling center sells 200 bowling balls per year It costs $4 to store one bowling ball for a year To reorder, there is a fixed cost of $1, plus $0.50 for each bowling ball How many times per year should the shop order bowling balls, and in what lot size, in order to minimize inventory costs?
Minimizing inventory costs. A retail outlet foe Boxowitz Calculators sells 720 calculators per year. It costs $2 to store one calculator for a year. To reorder, there is a fixed cost of $5, plus $2.50 each calculator. How many times per year should the store order calculators, and in what lot size, in order to minimize inventory costs?45. Minimizing inventory costs. Bon Temps Surf and Scuba Shop sells 360 surfboards per year. It costs $8 to store one surfboard for a year. Each reorder costs $10, plus an additional $5 for each surfboard ordered. How many times per year should the store order surfboards, and in what lot size, in order to minimize inventory costs?
Minimizing inventory costs. Repeat Exercise 44 using the same data, but assume yearly sales of 256 calculators with the fixed cost of each reorder set at $4.Minimizing inventory costs. Repeat Exercise 45 using the same data but change the reorder costs from an additional $5 per surfboard to $6 per surfboardMinimizing surface area. A closed-top cylindrical container is to have a volume of 250in2 what dimensions (radius and height) will minimize the surface area?Minimizing surface area. An open-top cylindrical container is to have a volume of 400cm2. what dimensions (radius and height) will minimize the surface area?50. Minimizing cost. Assume that the costs of the materials for making the cylindrical container described in Exercise 48 for the wall what dimensions will minimize the cost of materials?
Minimizing cost. Assume that the costs of the materials for making the cylindrical container described in Exercise 49 are0.0015/cm2forthebaseand0.008/cm2 for the wall. What dimensions will minimize the cost of materials?Maximizing volume. The postal service places a limit of 84 in. on the combined length and girth of (distance around) a package to be sent parcel post What dimensions of a rectangular box with square cross-section will contain the largest volume that can be mailed? (Hint There are two different girths.)53. Minimizing cost. A rectangular play area of is to be fenced off in a person’s yard The next-door neighbor agrees to pay half the cost of the fence on the side of the play area that lies along the property line What dimensions will minimize the cost of the fence?
54EMaximizing light. Repeat Exercises 54 but assume that the semicircle is to be stained glass, which transmits only half as much light as clear glass does.56-110. Use a spreadsheet to numerically verify the result of Exercises 1-55.
56. Of all numbers whose sum is 70, find the two that have the maximum product That is, maximize, .
56-110. Use a spreadsheet to numerically verify the result of Exercises 1-55.
57. Of all numbers whose sum is 50, find the two that have the maximum product That is, maximize, .
58E56-110. Use a spreadsheet to numerically verify the result of Exercises 1-55.
59. Of all numbers whose difference is 4, find the two that have the minimum product.
60E5 Use a spreadsheet to numerically verify the result of Exercises 1-55. Maximize Q=xy2 where x and y are positive numbers such that x+y2=162E63E5 Use a spreadsheet to numerically verify the result of Exercises 1-55. Maximum Q=xy, where x and y are positive numbers such that x+43y2=165E5 Use a spreadsheet to numerically verify the result of Exercises 1-55. Maximizing area. A rancher wants to enclose two rectangular areas near a river, one for sheep and one for cattle. There are 240 yd of fencing available. What is the largest total area that can be enclosed?5 Use a spreadsheet to numerically verify the result of Exercises 1-55. Maximizing area. A lifeguard needs to rope off a rectangular swimming area in front of Long Lake Beach, using 180 yd of rope and floats. What dimensions of the rectangle will maximize the area? What is the maximum area? (Note that the shoreline is one side of the rectangle.)5 Use a spreadsheet to numerically verify the result of Exercises 1-55. Maximizing area. Hentz Industries plans to enclose three parallel rectangular areas for sorting returned goods The three areas are within one large rectangular area and 1200 yd of fencing is available What is the largest total area that can be enclosed?56-110. Use a spreadsheet to numerically verify the result of Exercises 1-55.
69. Maximizing area. Grayson Farms plans to enclose three parallel rectangular livestock pens within one large rectangular area using 600 ft of fencing One side of the enclosure is a pre-existing stone wall.
a. If the three rectangular pens have their longer sides parallel to the stone wall, find the largest possible total area that can be enclosed
b. If the three rectangular pens have their shorter sides perpendicular to the stone wall, find the largest possible total area that can be enclosed
5 Use a spreadsheet to numerically verify the result of Exercises 1-55. Maximizing area. Of all rectangles that have a perimeter of 42 ft, find the dimensions of the one with the largest area. What is its area?71E5 Use a spreadsheet to numerically verify the result of Exercises 1-55. Maximizing volume. From a thin piece of cardboard 20 in, by 20 in., square comers are cut out so that the sides can be folded up to make a box What dimensions will yield a box of maximum volume? What is the maximum volume?56-110. Use a spreadsheet to numerically verify the result of Exercises 1-55.
73. Maximizing volume. From a 50-cm-by-50-cm sheet to aluminum, square corners are cut out so that the sides can be folded up to make a box. What dimensions will yield a box of maximum volume? What is the maximum volume?
56-110. Use a spreadsheet to numerically verify the result of Exercises 1-55.
74. Minimizing surface area. Mendoza Soup Company is constructing an open-top, square-based, rectangular metal tank that will have a volume of what dimensions will minimize surface area? What is the minimum surface area?
75E76E56-110. Use a spreadsheet to numerically verify the result of Exercises 1-55.
77. Minimizing surface area. Ever Green Gardening is designing a rectangular compost container that will be twice as tall as it is wide and must hold of composted food scraps. Find the dimensions of the compost container with minimal surface area (include the bottom and top)
78E79E80E56-110. Use a spreadsheet to numerically verify the result of Exercises 1-55.
For Exercises 23-28 find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit, Assume that revenue, Exercises 23-26
81.
82E83E84E56-110. Use a spreadsheet to numerically verify the result of Exercises 1-55.
For Exercises 23-28 find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit, Assume that revenue, Exercises 23-26
85. Maximizing profit. Raggs, Ltd., a clothing firm, determines that in order to sell x suits, the price per suit must be
It also determines that the total cost of producing x suits is given by
a. Find the total revenue,
b. Find the total profit,
c. How many suits must the company produce and sell in order to maximize profit?
d. What is the maximum profit?
e. What price per suit must be charged in order to maximize profit?
5 Use a spreadsheet to numerically verify the result of Exercises 1-55. For Exercises 23-28 find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit, Assume that revenue, R(x),andcost,C(x),areindollarsfor Exercises 23-26 Maximizing profit. Gritz-Charlston is a 300-unit luxury hotel All rooms are occupied when the hotel. Charges $80 per day for a room. For every increase of x dollars in the daily room rate, there are x rooms vacant Each occupied room costs $22 per day to service and maintain What should the hotel charge per day in order to maximize profit?87E56-110. Use a spreadsheet to numerically verify the result of Exercises 1-55.
For Exercises 23-28 find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit, Assume that revenue, Exercises 23-26
88. Maximizing parking tickets. Oak Glen currently employs 8 patrol officers who each write an average of 24 parking tickets per day For every additional officer placed on patrol, the average number of parking tickets per day written by each officer decreases by 4 How many additional officers should be placed on patrol in order to maximize the number of parking tickets written per day?
5 Use a spreadsheet to numerically verify the result of Exercises 1-55. For Exercises 23-28 find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit, Assume that revenue, R(x),andcost,C(x),areindollarsfor Exercises 23-26 Maximizing yield. Hood Apple Farm yields an average of 30 bushels of apples per tree when 20 trees are planted on an acre of ground. If 1 more tree is planted per acre, the yield decreases by 1 bushel (bu) per tree as a result of crowding. How many trees should be planted on an acre in order to get the highest yield?90E56-110. Use a spreadsheet to numerically verify the result of Exercises 1-55.
For Exercises 23-28 find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit, Assume that revenue, Exercises 23-26
91. Vanity license plates. According to a pricing model, increasing the fee for vanity license plates by $1 decreases the percentage of a state’s population that will request them by 0.04% (Source: E. D. Craft, “The demand for vanity (plates): Elasticities, net revenue maximization, and deadweight loss, “ Contemporary Economic Policy, )
a. Recently, the tee for vanity license plates in Maryland was $25, and the percentage of the state’s population that had vanity plates was 2 13% Use this information to construct the demand function, , for the percentage of Maryland’s population that will request vanity license plates for a fee of x dollars.
b. Find the fee, x, that will maximize revenue from vanity plates
92E5 Use a spreadsheet to numerically verify the result of Exercises 1-55. Minimizing costs. A rectangular box with a volume of 320ft3 is to be constructed with a square base and top The cost per square foot for the bottom is 15,forthetopis10,andforthesidesis2.5. What dimensions will minimize the cost?5 Use a spreadsheet to numerically verify the result of Exercises 1-55. Minimizing cost. A rectangular parking area measuring 5000ft2 is to be enclosed on three sides using chain-link fencing that costs $4.50 per foot The fourth side will be a wooden fence that costs $7 per foot What dimensions will minimize the total cost to enclose this area, and what is the minimum cost (rounded to the nearest dollar)?56-110. Use a spreadsheet to numerically verify the result of Exercises 1-55.
95. Minimizing cost. A rectangular garden measuring is to be enclosed on two parallel sides by stone wall that costs $35 per yd and the other two sides by wooden fencing that costs $28 per yd What dimensions will minimize the total cost of enclosing this garden, and what is the minimum cost (rounded to the nearest dollar)?
5 Use a spreadsheet to numerically verify the result of Exercises 1-55. Maximizing area. Bradley Publishing decides that each page in a new book must have an area of 73.125in2a0.75-in margin at the top and at the bottom of each page, and a 0.5-in margin on each of the sides. What should the outside dimensions of each page be so that the printed area is a maximum?5 Use a spreadsheet to numerically verify the result of Exercises 1-55. Minimizing inventory costs. A sporting goods store sells 100 pool tables per year It costs $20 to store one pool table for a year, To reorder, there is a fixed cost of $40 per shipment plus $16 for each pool table How many times per year should the store order pool tables, and in what lot size, in order to minimize inventory costs?56-110. Use a spreadsheet to numerically verify the result of Exercises 1-55.
98. Minimizing inventory costs. A pro shop in a bowling center sells 200 bowling balls per year It costs $4 to store one bowling ball for a year To reorder, there is a fixed cost of $1, plus $0.50 for each bowling ball How many times per year should the shop order bowling balls, and in what lot size, in order to minimize inventory costs?
56-110. Use a spreadsheet to numerically verify the result of Exercises 1-55.
99. Minimizing inventory costs. A retail outlet foe Boxowitz Calculators sells 720 calculators per year. It costs $2 to store one calculator for a year. To reorder, there is a fixed cost of $5, plus $2.50 each calculator. How many times per year should the store order calculators, and in what lot size, in order to minimize inventory costs?
5 Use a spreadsheet to numerically verify the result of Exercises 1-55. Minimizing inventory costs. Bon Temps Surf and Scuba Shop sells 360 surfboards per year. It costs $8 to store one surfboard for a year. Each reorder costs $10, plus an additional $5 for each surfboard ordered. How many times per year should the store order surfboards, and in what lot size, in order to minimize inventory costs?5 Use a spreadsheet to numerically verify the result of Exercises 1-55. Minimizing inventory costs. Repeat Exercise 44 using the same data, but assume yearly sales of 256 calculators with the fixed cost of each reorder set at $4.5 Use a spreadsheet to numerically verify the result of Exercises 1-55. Minimizing inventory costs. Repeat Exercise 45 using the same data but change the reorder costs from an additional $5 per surfboard to $6 per surfboard103E56-110. Use a spreadsheet to numerically verify the result of Exercises 1-55.
104. Minimizing surface area. An open-top cylindrical container is to have a volume of what dimensions (radius and height) will minimize the surface area?
56-110. Use a spreadsheet to numerically verify the result of Exercises 1-55.
For Exercises 23-28 find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit, Assume that revenue, Exercises 23-26
105. Minimizing cost. Assume that the costs of the materials for making the cylindrical container described in Exercise 48 for the wall what dimensions will minimize the cost of materials?
106E5 Use a spreadsheet to numerically verify the result of Exercises 1-55. Maximizing volume. The postal service places a limit of 84 in. on the combined length and girth of (distance around) a package to be sent parcel post What dimensions of a rectangular box with square cross-section will contain the largest volume that can be mailed? (Hint There are two different girths.)56-110. Use a spreadsheet to numerically verify the result of Exercises 1-55.
108. Minimizing cost. A rectangular play area of is to be fenced off in a person’s yard The next-door neighbor agrees to pay half the cost of the fence on the side of the play area that lies along the property line What dimensions will minimize the cost of the fence?
56-110. Use a spreadsheet to numerically verify the result of Exercises 1-55.
109. Maximizing light. A Norman window is a rectangle with a semicircle on top. Suppose that the perimeter of a particular Norman window is to be 24 ft. What should its dimensions be in order to allow the maximum amount of light to through the window?
5 Use a spreadsheet to numerically verify the result of Exercises 1-55. Maximizing light. Repeat Exercises 54 but assume that the semicircle is to be stained glass, which transmits only half as much light as clear glass does.For what positive number is the sum of its reciprocal and five times its square a minimum?For what positive number is the sum of its reciprocal and four times its square a minimum?113EA 24-in. piece of wire is cut in two pieces. One piece is used to form a circle and the other to form a square. How should the wire be cut so that the sum of the areas is a minimum?115. Business: minimizing costs. A power line is to be constructed from a power station at point A to an island at point C, which is 1 mi directly out in the water from a point B on the shore. Point B is 4 mi downshore from the power station at A. It costs $5000 per mile to lay the power line under water and $3000 per mile to lay the line under ground. At what point S downshore from A should the line come to the shore in order to minimize cost? Note that S could very well be B or A. (Hint: The length of CS is .) S is 3.25 mi downshore from A.
Life science: flights of homing pigeons. It is known that homing pigeons tend to avoid flying over water in the daytime, perhaps because downdrafts of air over water make flying difficult. Suppose a homing pigeon is released on an island at point C, which is 3 mi directly out in the water from a point B on shore. Point B is 8 mi downshore from the pigeons home loft at point A. Assume that a pigeon flying over water uses energy at a rate 1.28 times the rate over land. Toward what point S downshore from A should the pigeon fly in order to minimize the total energy required to get to the home loft at A? Assume that S is 4.245 mi downshore from A. Totalenergy=(Energyrateoverwater)(Distanceoverwater)+(Energyrateoverland)(Distanceoverland).117. A rectangular field is to be divided into two parallel rectangular areas, as shown in the figure below. If the total fencing available is k units, show that the length of each of the three parallel fences will be units.
Business: minimizing distance. A road is to be built between two cities C1 and C2, which are on opposite sides of a river of uniform width r. C1 is a units from the river, and C2 is b units from the river, with ab. A bridge will carry the traffic across the river. Where the bridge should be located in order to minimize the total distance between the cities? Give a general solution using the constants a, b, p, and r as shown in the figure. x=bpa+bBusiness: minimizing cost. The total cost, in dollars, of producing x units of a certain product is given by C(x)=8x+20+x3100. A(x)=8+20x+x2100 a. Find the average cost, A(x)=C(x)/x b. Find C(x)andA(x). c. Find the minimum of A(x) and the value x0 at which it occurs. Find C(x0). d. Compare A(x0)andC(x0).120E121. Minimize q, where are positive numbers, such that.
122. Minimize, where.
Business: minimizing inventory costsa general solution. A store sells Q units of a product per year. It costs a dollars to store one unit for a year. To reorder, there is a fixed cost of b dollars, plus c dollars for each unit. How many times per year should the store reorder, and in what lot size, in order to minimize inventory costs?124. Business: minimizing inventory costs. Use the general solution found in Exercise 123 to find how many times per year a store should reorder, and in what lot size, when
.
In Exercises 125–128, use a spreadsheet to maximize Q. Assume that all variables are positive, and state all answers to two decimal places.
125.
In Exercises 125–128, use a spreadsheet to maximize Q. Assume that all variables are positive, and state all answers to two decimal places.
126.
In Exercises 125–128, use a spreadsheet to maximize Q. Assume that all variables are positive, and state all answers to two decimal places.
127.
In Exercises 125128, use a spreadsheet to maximize Q. Assume that all variables are positive, and state all answers to two decimal places. Q=x2y2,wherex2+y2=501E2. Marginal revenue, cost, and profit, Let be, respectively, the revenue cost and profit in dollars from the production and sale of x items If
Find each of the following
a.
b.
c.
d.
e. Describe what each quantity in parts (b) and (d) represents
Marginal cost. Suppose the daily cost, in hundreds of dollars, of producing x security systems is C(x)=0.002x3+0.1x2+42x+300, and currently 40 security systems are produced daily. a. What is the current daily cost? b. What would be the additional daily cost of increasing production to 41 security systems daily? c. What is the marginal cost when x=40? d. Use marginal cost to estimate the daily cost of increasing production to 42 security systems daily.4. Marginal cost. Suppose the monthly cost, in dollars, of producing x daypacks is
And currently 25 daypacks are produced. Monthly
a. what is the current monthly cost?
b. what would be the additional cost of increasing production to 26 daypacks monthly?
c. what is the marginal cost when
d. Use marginal cost to estimate the difference in cost between producing 25 and 27 daypacks per month.
e. Use the answer from part (d) to predict
5E6EMarginal revenue. Solano Carriers finds that its monthly revenue, in dollars, from the sale of x carry-on suitcases is R(x)=0.007x30.5x2+150x. Currently Solano is selling 26 carry-on suitcases monthly. a. What is the current monthly revenue? b. How much would revenue increase if sales increased from 26 to 28 suitcases? c. What is the marginal revenue when 26 suitcases are sold? d. Use the answers from parts (a)-(c) to estimate the revenue resulting from selling 27 suitcases per month.8ESales. Let N(x) be the number of computers sold annually when the price is x dollars per computer Explain in words what occurs if N(100)=500,000andN(1000)=100. If the price increases from $1000 to $1001 sales will decrease by 100 units.10. Sales. Estimate the number of computers sold in Exercise 9 if the price of raised to $1025
497.500 computers.
For Exercise 11-16 assume that are in dollars and x is the number of units produced and sold
11. For the total-cost function
find .
For Exercise 11-16 assume that are in dollars and x is the number of units produced and sold
12. For the total-cost function
find
For Exercise 11-16 assume that are in dollars and x is the number of units produced and sold
13. For the total-revenue function
find
For Exercise 11-16 assume that C(x)andR(x) are in dollars and x is the number of units produced and sold For the total-revenue function R(x)=3x, find RandR(x)whenx=80andx=1For Exercise 11-16 assume that are in dollars and x is the number of units produced and sold
15. a. Using from Exercise 11 and from Exercise 14, find the total profit, .
b. Find
For Exercise 11-16 assume that are in dollars and x is the number of units produced and sold
16. a. Using from Exercise 12 and from Exercise 13, find the total profit, .
b. Find
Marginal supply. The supply S. of a new rollerball pen is given by S=0.007p30.5p2+150p, Where p is the price in dollars a. Find the rate of change of quantity with respect to price, dS/dp, b. How many units will producers want to supply when the price is $25 per unit? c. c) Find the rate of change at P = 25 and interpret this result d. d) Would you expect dS/dp to be positive or negative? Why?18. Average cost. The average cost for Turtlehead, Inc, to produce x units of its thermal outerwear is given by the function
Use A’ (x) to estimate the change in average cost as production goes from 100 units to 101 units.
19. Marginal productivity. An employee’s monthly productivity, M, in number of units produced, is found to be a function of the number of years of service, t For a certain product, the productivity function is given by
a. Find the productivity of an employee after 5 yr, 10 yr, 25 yr, and 45 yr, of service
b. Find the marginal productivity.
c. c) Find the marginal productivity at and interpret the results
d. d) Explain how an employee’s marginal productivity might be related to experience and to age.
20. Supply. A supply function for a certain product is given by
Where S (p) is the number of items produced when the price is p dollars Use to estimate how many more units a producer will supply when the price changes from $18.00 per unit to $18.20 per unit 31.95, or about 32 units.
Gross domestic product. The U.S gross domestic product, in billions of current dollars may be modeled by the function P(x)=567+x(36x0.6104), Where x is the number of years since 1960 (Source U S Bureau for Economic Analysis) Use P(x) to estimate how much the gross domestic product increased from 2014 to 2015.22. Advertising. Norris Inc. finds that it sells N units of a product after spending x thousands of dollars on advertising, where
Use to estimate how many mire units Norris will sell by increasing its advertising expenditure from $100,00 to $101,000.
Businesses and individuals are frequently concerned about their marginal tax rate, or the rate at which the next dollar earned is taxed In progressive taxation, the 80,001 st dollar earned is taxed at a higher rate than the 25.001 st dollar earned and at a lower rate than the 140,001 st dollar earned Use the following graph, showing the marginal tax rate for 2014 for single filers, to answer Exercises 23-26.
Single Filers Income Bracket Rate
0–9075 10%
9076–36900 15%
36901–89350 25%
89351–186350 28%
186351–405100 33%
405101–406750 35%
406751+ 39.6%
23. Was the taxation in 2014 progressive? Why or why not?
Businesses and individuals are frequently concerned about their marginal tax rate, or the rate at which the next dollar earned is taxed In progressive taxation, the 80,001 st dollar earned is taxed at a higher rate than the 25.001 st dollar earned and at a lower rate than the 140,001 st dollar earned Use the following graph, showing the marginal tax rate for 2014 for single filers, to answer Exercises 23-26.
Single Filers Income Bracket Rate
0–9075 10%
9076–36900 15%
36901–89350 25%
89351–186350 28%
186351–405100 33%
405101–406750 35%
406751+ 39.6%
24. Marcy and Tyrone work for the same marketing agency Because she is not yet a partner Marcy’s year-end income is approximately $95,000, Tyrone’s year-end income is approximately $185,000 Suppose one of them is to receive another $5000, 000, in income for the year Which one would keep more of that $5000 after taxes ? Why?
Businesses and individuals are frequently concerned about their marginal tax rate, or the rate at which the next dollar earned is taxed In progressive taxation, the 80,001 st dollar earned is taxed at a higher rate than the 25.001 st dollar earned and at a lower rate than the 140,001 st dollar earned Use the following graph, showing the marginal tax rate for 2014 for single filers, to answer Exercises 23-26.
Single Filers Income Bracket Rate
0–9075 10%
9076–36900 15%
36901–89350 25%
89351–186350 28%
186351–405100 33%
405101–406750 35%
406751+ 39.6%
25. Alan earns $85,000 per year and is considering a second job that would earn him another $10,000 annually At what rate will his tax liability (the amount he must pay in taxes) change if he takes the extra job? Express your answer in tax dollars paid per dollar earned.
Businesses and individuals are frequently concerned about their marginal tax rate, or the rate at which the next dollar earned is taxed In progressive taxation, the 80,001 st dollar earned is taxed at a higher rate than the 25.001 st dollar earned and at a lower rate than the 140,001 st dollar earned Use the following graph, showing the marginal tax rate for 2014 for single filers, to answer Exercises 23-26. Single Filers Income Bracket Rate 09075 10% 907636900 15% 3690189350 25% 89351186350 28% 186351405100 33% 405101406750 35% 406751+ 39.6% Iris earns $50,000 per year and is considering extra work that would earn her an extra $3000 annually At what rate will her tax liability grow if she takes the extra work (See Exercise 25)?Find yandf(x)x Round to four and two decimal places, respectively Fory=f(x)=x3,x=2andx=0.01Find Round to four and two decimal places, respectively
28.
Find yandf(x)x Round to four and two decimal places, respectively Fory=f(x)=x+x2,x=3andx=0.04Find Round to four and two decimal places, respectively
30.
Find Round to four and two decimal places, respectively
31.
Find yandf(x)x Round to four and two decimal places, respectively Fory=f(x)=1/x2,x=1,andx=0.5Find yandf(x)x Round to four and two decimal places, respectively Fory=f(x)=3x1,x=4andx=2Find yandf(x)x Round to four and two decimal places, respectively Fory=f(x)=2x3,x=8,andx=0.5Use to find a decimal approximation of each radical expression Round to three decimal places.
35.
Use yf(x)x to find a decimal approximation of each radical expression Round to three decimal places. 8Use yf(x)x to find a decimal approximation of each radical expression Round to three decimal places. 102Use to find a decimal approximation of each radical expression Round to three decimal places.
38.
Use yf(x)x to find a decimal approximation of each radical expression Round to three decimal places. 10053Use yf(x)x to find a decimal approximation of each radical expression Round to three decimal places. 283Find .
41.
Find.
42.
Find.
43.
Find.
44.
Find.
45.
Find .
46.
Find dydx. y=x42x3+5x2+3x4Find dydx. y=(7x)8In Exercise 48 ,finddywhenx=1anddx=0.01.In Exercise 47, finddywhenx=2anddx=0.1.51. For .
52. For .
53. For .
54. For .
55EMedical dosage. The function N(t)=0.8t+10005t+4 gives the bodily concentration N(t), in parts per million, of a dosage of medication after t hours Use differentials to determine whether the concentration changes more from 1 0 hr to 1.1 hr or from 2.8 hr to 2.9 hr.Major League ticket prices. The average ticket price of a major league baseball game can be modeled by the function P(x)=0.06x30.5x2+1.64x+24.76, Where x is the number of years after 2008 (Source Major League Baseball ) Use differentials to predict whether ticket price will increase more between 2010 and 2012 or between 2014 and 2016 Ticket price will increase more between 2014 and 2016.Suppose a rope surrounds the earth at the equator. The rope is lengthened by 10 ft By about how much is the rope raised above the earth?Marginal average cost. In Section 1.6 we defined the average cost of producing x units of a product in terms of the total cost C(x)=byA(x)=C(x)/x Find a general expression for marginal average cost A(x).60. Cost and tolerance. A firm contracts to paint the exterior of a large water tank in the shape of a half-dome (a hemisphere) The radius of the tank is measured to be 100 ft with a tolerance of (The formula for the surface area of a hemisphere is as an approximation for ) Each can of paint costs $30 and covers
a. Calculate the approximate difference in the surface area due to the tolerance
b. Assuming the painters cannot bring partial cans of paint to the job, how many extra cans should they bring to cover the extra area they may encounter?
c. How much extra should the painters plan to spend on paint to account for the possible extra area?
61. Strategic oil supply. The U.S Strategic Petroleum Reserve (SPR) stores petroleum in large spherical caverns built into salt deposits along the Gulf of Mexico (Source, U.S. Department of Energy.) These caverns can be enlarged by filling the void with water, which dissolves the surrounding salt, and then pumping brine out Suppose a cavern has a radius of 400 ft, which engineers want to enlarge by 2 ft. Use a differential to estimate how much volume will be added to form the enlarged cavern (The formula for the volume of a sphere is use 3, 14 as an approximation for)
In each of Exercises 62-66, a demand function, p=D(x) expresses price, in dollars, as a function of the number of items produced and sold. Find the marginal revenue p=100xIn each of Exercises 62-66, a demand function, p=D(x) expresses price, in dollars, as a function of the number of items produced and sold. Find the marginal revenue p=400xIn each of Exercises 62-66, a demand function, expresses price, in dollars, as a function of the number of items produced and sold. Find the marginal revenue
64.
In each of Exercises 62-66, a demand function, p=D(x) expresses price, in dollars, as a function of the number of items produced and sold. Find the marginal revenue p=4000x+3In each of Exercises 62-66, a demand function, p=D(x) expresses price, in dollars, as a function of the number of items produced and sold. Find the marginal revenue. p=3000x+5Look up differential in a book or Wed site devoted to math history In a short paragraph, describe your findings.68. Explain the uses of the differential.
For the demand function given in each of Exercises 1-10, find the following a. The elasticity b. The elasticity at the given price, stating whether the demand is elastic or inelastic c. The value (s) of x for which total revenue is a maximum (assume that x is in dollars) q=D(x)=400x;x=125For the demand function given in each of Exercises 1-10, find the following a. The elasticity b. The elasticity at the given price, stating whether the demand is elastic or inelastic c. The value (s) of x for which total revenue is a maximum (assume that x is in dollars) q=D(x)=500x;x=38For the demand function given in each of Exercises 1-10, find the following a. The elasticity b. The elasticity at the given price, stating whether the demand is elastic or inelastic c. The value (s) of x for which total revenue is a maximum (assume that x is in dollars) q=D(x)=2004x;x=46For the demand function given in each of Exercises 1-10, find the following a. The elasticity b. The elasticity at the given price, stating whether the demand is elastic or inelastic c. The value (s) of x for which total revenue is a maximum (assume that x is in dollars) q=D(x)=5002x;x=57For the demand function given in each of Exercises 1-10, find the following
a. The elasticity
b. The elasticity at the given price, stating whether the demand is elastic or inelastic
c. The value (s) of x for which total revenue is a maximum (assume that x is in dollars)
5.
For the demand function given in each of Exercises 1-10, find the following
a. The elasticity
b. The elasticity at the given price, stating whether the demand is elastic or inelastic
c. The value (s) of x for which total revenue is a maximum (assume that x is in dollars)
6.
For the demand function given in each of Exercises 1-10, find the following
a. The elasticity
b. The elasticity at the given price, stating whether the demand is elastic or inelastic
c. The value (s) of x for which total revenue is a maximum (assume that x is in dollars)
7.
For the demand function given in each of Exercises 1-10, find the following
a. The elasticity
b. The elasticity at the given price, stating whether the demand is elastic or inelastic
c. The value (s) of x for which total revenue is a maximum (assume that x is in dollars)
8.
For the demand function given in each of Exercises 1-10, find the following a. The elasticity b. The elasticity at the given price, stating whether the demand is elastic or inelastic c. The value (s) of x for which total revenue is a maximum (assume that x is in dollars) q=D(x)=100(x+3)2;x=1For the demand function given in each of Exercises 1-10, find the following a. The elasticity b. The elasticity at the given price, stating whether the demand is elastic or inelastic c. The value (s) of x for which total revenue is a maximum (assume that x is in dollars) q=D(x)=500(2x+12)2;x=8Demand for oil. Suppose you have been hired as an economic consultant concerning the world demand for oil. The demand function is q=D(x)=50,000+300x3x2,for0x180, Where q is measured in millions of barrels of oil per day at a price of x dollars per barrel a. Find the elasticity E(x)=6x2300x50,000+300x3x2 b. Find the elasticity at a price of $75 per barrel, stating whether demand is elastic or inelastic at that price c. Find the elasticity at a price of $100 per barrel stating whether demand is elastic or inelastic at that price d. Find the elasticity at a price of $20 per barrel, stating whether demand is elastic or inelastic at that price e. At what price is revenue a maximum? f. What quantity of oil will be sold at the price that maximizes revenue? Compare the current world price to your answer g. At a price of $110 per barrel, will a small increase in price cause total revenue to increase or decrease?Demand for chocolate chip cookies. Good Times Bakers determines that the demand function for its chocolate chip cookies is q=D(x)=96725x, Where q is the quantity of cookies sold when the price is x cents per cookie. a. Find the elasticity E(x)=25x96725x b. At what price is elasticity of demand equal to 1? c. At what prices is elasticity of demand elastic? d. At what prices is elasticity of demand inelastic? e. At what price is the revenue a maximum? f. At a price of 20 per cookie, will a small increase in price cause total revenue to increase or decrease?13. Demand for computer games. High Wire Electronics determines the following demand function for a new game:
Where q is the number of games sold per day when the price is x dollars per game.
a. Find the elasticity
b. Find the elasticity when
c. . Will a small increase in price cause total revenue to increase or decrease?
14. Demand for tomato plants. Sunshine Gardens determines the following demand function during early summer for tomato plants
Where q is the number of plants sold per day when the price is x dollars per plant
a. Find the elasticity
b. Find the elasticity when
c. At $3 per plant, will a small increase in price cause total revenue to increase or decrease?
15. Business. Tipton Industries determines that the demand function for its sunglass cases is
Where q is the quantity of sunglass cases sold when the price is x dollars per case
a. Find the elasticity of demand
b. Find the elasticity when
c. At what price is revenue a maximum?
d. If the price is $8 and Tipton Industries raises the price by 20%, will revenue increase or decrease?
Does this contradict the answers to parts (b) and (c)? Why or why not?
Economics: constant elasticity curve. a. Find the elasticity of the demand function q=D(x)=kxn, Where k is a positive constant and n is an integer greater than 0. b. Is the value of the elasticity dependent on the price per unit? c. For what value of n does total revenue have a maximum?Differentiate implicitly to find dy/dx. Then find the slope of the curve at the given point 3x3y2=8;(2,4)Differentiate implicitly to find. Then find the slope of the curve at the given point
2.
Differentiate implicitly to find. Then find the slope of the curve at the given point
3.
Differentiate implicitly to find dy/dx. Then find the slope of the curve at the given point 2x23y3=5;(2,1)Differentiate implicitly to find. Then find the slope of the curve at the given point
5.
Differentiate implicitly to find. Then find the slope of the curve at the given point
6.
Differentiate implicitly to find. Then find the slope of the curve at the given point
7.
Differentiate implicitly to find dy/dx. Then find the slope of the curve at the given point 3x2y4=12;(2,1)Differentiate implicitly to find dy/dx. Then find the slope of the curve at the given point x4x2y3=12;(2,1)Differentiate implicitly to find dy/dx. Then find the slope of the curve at the given point x3x2y2=9;(3,2)Differentiate implicitly to find dy/dx. Then find the slope of the curve at the given point xy+y22x=0;(1,2)Differentiate implicitly to find dy/dx. Then find the slope of the curve at the given point xyx+2y=3;(5,23)Differentiate implicitly to find dy/dx. Then find the slope of the curve at the given point 4x3y43y+5x+1=0;(1,2)Differentiate implicitly to find. Then find the slope of the curve at the given point
14.
Differentiate implicitly to find
15.
Differentiate implicitly to find dy/dx. 2xy+3=0Differentiate implicitly to find dy/dx. x2y2=16Differentiate implicitly to find
18.
Differentiate implicitly to find
19.
Differentiate implicitly to find
20.
Differentiate implicitly to find dy/dx. x2y3+x3y4=11Differentiate implicitly to find
22.
For each demand equation in Exercises 23-30, differentiate implicitly to.
23.
For each demand equation in Exercises 23-30, differentiate implicitly to.
24.
For each demand equation in Exercises 23-30, differentiate implicitly to dp/dx. xp3=24For each demand equation in Exercises 23-30, differentiate implicitly to.
26.
For each demand equation in Exercises 23-30, differentiate implicitly to dp/dx. x2p+xp+12x+p=1 (Hint: Clear the fraction first.)For each demand equation in Exercises 23-30, differentiate implicitly to.
28. (Hint: Clear the fraction first.)
For each demand equation in Exercises 23-30, differentiate implicitly to dp/dx. (p+4)(x+3)=48For each demand equation in Exercises 23-30, differentiate implicitly to.
30.
Nonnegative variable quantities G and H are related by the equation G2+H2=25. What is the rate of change dH/dtwhendG/dt=3andG=0?G=1?G=3?32. Variable quantities A and B are related by the equation
.
What is the rate of changeat the moment when
In Exercises 33-36, find the rates of change of total revenue, cost, and profit with respect to time. Assume that R(x)andC(x) are in dollars. R(x)=50x0.5x2 C(x)=10x+3, whenx=10anddx/dt=5unitsperdayIn Exercises 33-36, find the rates of change of total revenue, cost, and profit with respect to time. Assume that are in dollars.
34. ,
when units per day
In Exercises 33-36, find the rates of change of total revenue, cost, and profit with respect to time. Assume that R(x)andC(x) are in dollars. R(x)=280x0.4x2, C(x)=5000+0.6x2, whenx=200anddx/dt=300unitsperdayIn Exercises 33-36, find the rates of change of total revenue, cost, and profit with respect to time. Assume that are in dollars.
36. ,
,
37. Change of sales. Suppose that the price p, in dollars, and number of sales, x, of a mechanical pencil are related by
.
If p and x are both functions of time, measured in days, find the rate at which x is changing when
Change of revenue. For x and p as described in Exercise 37, find the rate at which total revenue R=xp is changing when x=3,p=5,anddpdt=1.5.Rate of change of the Arctic ice cap. In a trend that scientists attribute, at least in part, to global warming, the floating cap of sea ice on the Arctic Ocean has been shrinking since 1980. The ice cap always shrinks in summer and grows in winter. Average minimum size of the ice cap, in square miles, can be approximated by A=r2. In 2013, the radius of the ice cap was approximately 792 mi and was shrinking at a rate of approximately 4.7 mi/yr. (Source: Based on data from nsidc.org.) How fast was the area changing at that time?40E41. Body surface area. Certain chemotherapy dosages depend on a patient’s surface area. According to the Mosteller model,
,
where h is the patient’s height in centimeters, w is the patient’s weight in kilograms, and S is the approximation of the patient’s surface area in square meters. (Source: www.halls.md.) Assume that Kim’s height is a constant 165 cm, but she is losing weight. If she loses 2 kg per month, how fast is her surface area decreasing at the instant she weighs 70 kg?
42E