Precalculus Enhanced with Graphing Utilities
Precalculus Enhanced with Graphing Utilities
6th Edition
ISBN: 9780321795465
Author: Michael Sullivan, Michael III Sullivan
Publisher: PEARSON
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Chapter 3.1, Problem 52AYU

Developing a Linear Model from Data The following data represent the various combinations of soda and hot dogs that Yolanda can buy at a baseball game with $ 60 .

Chapter 3.1, Problem 52AYU, Developing a Linear Model from Data The following data represent the various combinations of soda

a. Plot the ordered pairs ( s , h ) in a Cartesian plane.

b. Show that the number of hot dogs purchased h is a linear function of the number of sodas purchased s .

c. Determine the linear function that describes the relation between s and h .

d. What is the implied domain of the linear function?

e. Graph the linear function in the Cartesian plane drawn in part (a).

f. Interpret the slope.

g. Interpret the values of the intercepts.

Expert Solution & Answer
Check Mark
To determine

To calculate:

a. Plot the ordered pairs ( s,h ) in a Cartesian plane.

b. Show that the number of hot dogs purchased h is a linear function of the number of sodas purchased s .

c. Determine the linear function that describes the relation between s and h .

d. What is the implied domain of the linear function?

e. Graph the linear function in the Cartesian plane drawn in part (a..

f. Interpret the slope.

g. Interpret the values of the intercepts.

Answer to Problem 52AYU

Solution:

a. The plot of the data in Cartesian plane is as shown below:

Precalculus Enhanced with Graphing Utilities, Chapter 3.1, Problem 52AYU , additional homework tip  1

b. Yes, it is proved that the number of hot dogs purchased h is a linear function of the number of sodas purchased s .

c. The linear function of the given data is y = 0.6x + 12 .

d. The implied domain of the given function is (, ) .

e. The graph is as shown below:

Precalculus Enhanced with Graphing Utilities, Chapter 3.1, Problem 52AYU , additional homework tip  2

f. The function is an decreasing function.

g. Therefore, the y-intercept is 12 and x-intercept is 20.

Explanation of Solution

Given:

The given data is

Precalculus Enhanced with Graphing Utilities, Chapter 3.1, Problem 52AYU , additional homework tip  3

Formula Used:

Average rate of change if a function f(x) from a to b is m= f(b)  f(a) b  a .

Point slope formula: y   y 1   = m(x   x 1 )

Calculation:

a. The plot of the given data in a Cartesian plane is

Precalculus Enhanced with Graphing Utilities, Chapter 3.1, Problem 52AYU , additional homework tip  4

b. Here, in the above Cartesian plane, we can see that the points are arranged in a straight line format.

On joining the given points, we get a straight non vertical line.

Therefore, we can say that the given number of hot dogs purchased h is a linear function of number of sodas purchased s .

c. From the given data, we can find the average rate of change.

When

a = 20

b = 15

f(a) = 0

f(b) = 3

Now, we get the average rate of change of the given function as

m =  3  0 15  20

=  3 5  = 0.6

Thus, the rate of change of the given function is 0.6 .

Now, we have to use this in the point slope formula, where x 1  = 20 and y 1  = 0 .

Thus, we get

y   y 1  = m(x   x 1 )

 y  0 = 0.6(x  20)

 y = 0.6x + 12

Therefore, the linear function of the given data is y = 0.6x + 12 .

d. We can see that the value of x in the linear function can be any number. Therefore, the implied domain of the given function is (, ) .

e. The graph of the linear function is

Precalculus Enhanced with Graphing Utilities, Chapter 3.1, Problem 52AYU , additional homework tip  5

f. We know that the average rate of change of the function is itself its slope.

Thus, here we have seen that the slope is 0.6 which is a negative number.

Therefore, the function is an decreasing function.

g. x-intercept is to be calculated when x is equal to zero and x intercept is to be calculated when y is equal to zero.

From the function y = 0.6x + 12 , we can calculate the x intercepts and y intercept.

x-intercept :

y = 0.6x + 12

0 = 0.6x + 12

x =  12 0.6

x = 20

y-intercept :

y = 0.6(0) + 12

y = 12

Therefore, the y-intercept is 12 and x-intercept is 20.

Chapter 3 Solutions

Precalculus Enhanced with Graphing Utilities

Ch. 3.1 - What is the only type of function that has a...Ch. 3.1 - A car has 12,500 miles on its odometer. Say the...Ch. 3.1 - In Problems 13-20, a linear function is given. a....Ch. 3.1 - In Problems 13-20, a linear function is given. a....Ch. 3.1 - In Problems 13-20, a linear function is given. a....Ch. 3.1 - In Problems 13-20, a linear function is given. a....Ch. 3.1 - In Problems 13-20, a linear function is given. a....Ch. 3.1 - In Problems 13-20, a linear function is given. a....Ch. 3.1 - In Problems 13-20, a linear function is given. a....Ch. 3.1 - In Problems 13-20, a linear function is given. a....Ch. 3.1 - In Problems 21-28, determine whether the given...Ch. 3.1 - In Problems 21-28, determine whether the given...Ch. 3.1 - In Problems 21-28, determine whether the given...Ch. 3.1 - In Problems 21-28, determine whether the given...Ch. 3.1 - In Problems 21-28, determine whether the given...Ch. 3.1 - In Problems 21-28, determine whether the given...Ch. 3.1 - In Problems 21-28, determine whether the given...Ch. 3.1 - In Problems 21-28, determine whether the given...Ch. 3.1 - Suppose that f( x )=4x1 and g(x)=2x+5 . a. Solve...Ch. 3.1 - Suppose that f( x )=3x+5 and g(x)=2x+15 . a. Solve...Ch. 3.1 - In parts (a) - (f), use the following figure. a....Ch. 3.1 - In parts (a) - (f), use the following figure. a....Ch. 3.1 - In parts (a) and (b), use the following figure. a....Ch. 3.1 - In parts (a) and (b), use the following figure. a....Ch. 3.1 - In parts (a) and (b), use the following figure. a....Ch. 3.1 - In parts (a) and (b), use the following figure. a....Ch. 3.1 - Car Rentals The cost C , in dollars, of a one-day...Ch. 3.1 - Phone Charges The monthly cost C , in dollars, for...Ch. 3.1 - Supply and Demand Suppose that the quantity...Ch. 3.1 - Supply and Demand Suppose that the quantity...Ch. 3.1 - Taxes The function T( x )=0.15(x9225)+922.50...Ch. 3.1 - Competitive Balance Tax In 2011, major league...Ch. 3.1 - The point at which a company’s profits equal...Ch. 3.1 - The point at which a company’s profits equal...Ch. 3.1 - Straight-line Depreciation Suppose that a company...Ch. 3.1 - Straight-line Depreciation Suppose that a company...Ch. 3.1 - Cost Function The simplest cost function is the...Ch. 3.1 - Cost Function Refer to Problem 47. 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(b)...Ch. 3.2 - In Problems 11-16, (a) Draw a scatter diagram. (b)...Ch. 3.2 - In Problems 11-16, (a) Draw a scatter diagram. (b)...Ch. 3.2 - In Problems 11-16, (a) Draw a scatter diagram. (b)...Ch. 3.2 - In Problems 11-16, (a) Draw a scatter diagram. (b)...Ch. 3.2 - In Problems 11-16, (a) Draw a scatter diagram. (b)...Ch. 3.2 - Prob. 17AYUCh. 3.2 - Prob. 18AYUCh. 3.2 - Prob. 19AYUCh. 3.2 - Prob. 20AYUCh. 3.2 - Prob. 21AYUCh. 3.2 - Prob. 22AYUCh. 3.2 - Prob. 23AYUCh. 3.2 - Prob. 24AYUCh. 3.2 - Prob. 25AYUCh. 3.3 - List the intercepts of the equation y= x 2 9 ....Ch. 3.3 - Prob. 2AYUCh. 3.3 - To complete the square of x 2 5x , you add the...Ch. 3.3 - To graph y= (x4) 2 you shift the graph of y= x 2...Ch. 3.3 - The graph of a quadratic function is called a(n)...Ch. 3.3 - The vertical line passing through the vertex of a...Ch. 3.3 - The x-coordinate of the vertex of f( x )=a x 2...Ch. 3.3 - True or False The graph of f( x )=2 x 2 +3x4 opens...Ch. 3.3 - True or False The y-coordinate of the vertex of f(...Ch. 3.3 - True or False If the discriminant b 2 4ac=0 , the...Ch. 3.3 - If b 2 4ac0 , which of the following conclusions...Ch. 3.3 - If the graph of f( x )=a x 2 +bx+c,a0 , has a...Ch. 3.3 - In Problems 13-20, match each graph to one the...Ch. 3.3 - In Problems 13-20, match each graph to one the...Ch. 3.3 - In Problems 13-20, match each graph to one the...Ch. 3.3 - In Problems 13-20, match each graph to one the...Ch. 3.3 - In Problems 13-20, match each graph to one the...Ch. 3.3 - In Problems 13-20, match each graph to one the...Ch. 3.3 - In Problems 13-20, match each graph to one the...Ch. 3.3 - In Problems 13-20, match each graph to one the...Ch. 3.3 - In Problems 21-32, graph the function f by...Ch. 3.3 - In Problems 21-32, graph the function f by...Ch. 3.3 - In Problems 21-32, graph the function f by...Ch. 3.3 - In Problems 21-32, graph the function f by...Ch. 3.3 - In Problems 21-32, graph the function f by...Ch. 3.3 - In Problems 21-32, graph the function f by...Ch. 3.3 - In Problems 21-32, graph the function f by...Ch. 3.3 - In Problems 21-32, graph the function f by...Ch. 3.3 - In Problems 21-32, graph the function f by...Ch. 3.3 - In Problems 21-32, graph the function f by...Ch. 3.3 - In Problems 21-32, graph the function f by...Ch. 3.3 - In Problems 21-32, graph the function f by...Ch. 3.3 - 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In Problems 63-74, (a) graph each function, (b)...Ch. 3.3 - In Problems 63-74, (a) graph each function, (b)...Ch. 3.3 - In Problems 63-74, (a) graph each function, (b)...Ch. 3.3 - In Problems 63-74, (a) graph each function, (b)...Ch. 3.3 - In Problems 63-74, (a) graph each function, (b)...Ch. 3.3 - In Problems 63-74, (a) graph each function, (b)...Ch. 3.3 - In Problems 63-74, (a) graph each function, (b)...Ch. 3.3 - The graph of the function f( x )=a x 2 +bx+c has...Ch. 3.3 - The graph of the function f(x)=a x 2 +bx+c has...Ch. 3.3 - In Problems 77-82, for the given functions fandg ,...Ch. 3.3 - In Problems 77-82, for the given functions fandg ,...Ch. 3.3 - In Problems 77-82, for the given functions fandg ,...Ch. 3.3 - In Problems 77-82, for the given functions fandg ,...Ch. 3.3 - In Problems 77-82, for the given functions fandg ,...Ch. 3.3 - In Problems 77-82, for the given functions fandg ,...Ch. 3.3 - Answer Problems 83 and 84 using the following: A...Ch. 3.3 - Answer Problems 83 and 84 using the following: A...Ch. 3.3 - Suppose that f(x)= x 2 +4x21 . (a) What is the...Ch. 3.3 - Suppose that f( x )= x 2 +2x8 . (a) What is the...Ch. 3.3 - Find the point on the line y=x that is closest to...Ch. 3.3 - Find the point on the line y=x+1 that is closest...Ch. 3.3 - Maximizing Revenue Suppose that the manufacturer...Ch. 3.3 - Maximizing Revenue The John Deere company has...Ch. 3.3 - Minimizing Marginal Cost The marginal cost of a...Ch. 3.3 - Minimizing Marginal Cost (See Problem 91.) The...Ch. 3.3 - Business The monthly revenue R achieved by selling...Ch. 3.3 - Business The daily revenue R achieved by selling x...Ch. 3.3 - Stopping Distance An accepted relationship between...Ch. 3.3 - Birthrate for Unmarried Women In the United...Ch. 3.3 - Let f( x )=a x 2 +bx+c , where a,b,andc are odd...Ch. 3.3 - Make up a quadratic function that opens down and...Ch. 3.3 - On one set of coordinate axes, graph the family of...Ch. 3.3 - On one set of coordinate axes, graph the family of...Ch. 3.3 - State the circumstances that cause the graph of a...Ch. 3.3 - Why does the graph of a quadratic function open up...Ch. 3.3 - Can a quadratic function have a range of ( , ) ?...Ch. 3.3 - What are the possibilities for the number of times...Ch. 3.4 - Translate the following sentence into a...Ch. 3.4 - Use a graphing utility to find the line of best...Ch. 3.4 - Maximizing Revenue The price p (in dollars) and...Ch. 3.4 - Maximizing Revenue The price p (in dollars) and...Ch. 3.4 - Maximizing Revenue The price p (in dollars) and...Ch. 3.4 - Maximizing Revenue The price p (in dollars) and...Ch. 3.4 - Enclosing a Rectangular Field David has 400 yards...Ch. 3.4 - Enclosing a Rectangular Field Beth has 3000 feet...Ch. 3.4 - Enclosing a Rectangular Field with a Fence A...Ch. 3.4 - Enclosing a Rectangular Field with a Fence A...Ch. 3.4 - Analyzing the Motion of a Projectile A projectile...Ch. 3.4 - Analyzing the Motion of a Projectile A projectile...Ch. 3.4 - Suspension Bridge A suspension bridge with weight...Ch. 3.4 - Architecture A parabolic arch has a span of 120...Ch. 3.4 - Constructing Rain Gutters A rain gutter is to be...Ch. 3.4 - Norman Windows A Norman window has the shape of a...Ch. 3.4 - Constructing a Stadium A track-and-field playing...Ch. 3.4 - Architecture A special window has the shape of a...Ch. 3.4 - Chemical Reactions A self-catalytic chemical...Ch. 3.4 - Calculus: Simpson's Rule The figure shows the...Ch. 3.4 - Use the result obtained in Problem 20 to find the...Ch. 3.4 - Use the result obtained in Problem 20 to find the...Ch. 3.4 - Use the result obtained in Problem 20 to find the...Ch. 3.4 - Use the result obtained in Problem 20 to find the...Ch. 3.4 - Life Cycle Hypothesis An individuals income varies...Ch. 3.4 - Height of a Rail A shot-putter throws a hall at an...Ch. 3.4 - Which Model? The following data represent the...Ch. 3.4 - Which Model? An engineer collects the following...Ch. 3.4 - Which Model? The following data represent the...Ch. 3.4 - Which Model? A cricket makes a chirping noise by...Ch. 3.4 - Refer to Example 1 in this section. Notice that if...Ch. 3.5 - Solve the inequality 3x27 .Ch. 3.5 - Write (2,7] using inequality notation.Ch. 3.5 - (a) f( x )0 (b) f( x )0Ch. 3.5 - (a) g( x )0 (b) g( x )0Ch. 3.5 - (a) g( x )f( x ) (b) f( x )g( x )Ch. 3.5 - (a) f( x )g( x ) (b) f( x )g( x )Ch. 3.5 - x 2 3x100Ch. 3.5 - x 2 +3x100Ch. 3.5 - x 2 4x0Ch. 3.5 - x 2 +8x0Ch. 3.5 - x 2 90Ch. 3.5 - x 2 10Ch. 3.5 - x 2 +x12Ch. 3.5 - x 2 +7x12Ch. 3.5 - 2 x 2 5x+3Ch. 3.5 - 6 x 2 6+5xCh. 3.5 - x 2 x+10Ch. 3.5 - x 2 +2x+40Ch. 3.5 - 4 x 2 +96xCh. 3.5 - 25 x 2 +1640xCh. 3.5 - 6( x 2 1 )5xCh. 3.5 - 2( 2 x 2 3x )9Ch. 3.5 - What is the domain of the function f( x )= x 2 16...Ch. 3.5 - What is the domain of the function f( x )= x3 x 2...Ch. 3.5 - In Problems 25-32, use the given functions f and g...Ch. 3.5 - In Problems 25-32, use the given functions f and g...Ch. 3.5 - In Problems 25-32, use the given functions f and g...Ch. 3.5 - In Problems 25-32, use the given functions f and g...Ch. 3.5 - In Problems 25-32, use the given functions f and g...Ch. 3.5 - In Problems 25-32, use the given functions f and g...Ch. 3.5 - In Problems 25-32, use the given functions f and g...Ch. 3.5 - In Problems 25-32, use the given functions f and g...Ch. 3.5 - Physics A ball is thrown vertically upward with an...Ch. 3.5 - Physics A ball is thrown vertically upward with an...Ch. 3.5 - Revenue Suppose that the manufacturer of a gas...Ch. 3.5 - Revenue The John Deere company has found that the...Ch. 3.5 - Artillery A projectile Fired from the point ( 0,0...Ch. 3.5 - Runaway Car Using Hooke's Law, we can show that...Ch. 3.5 - Show that the inequality ( x4 ) 2 0 has exactly...Ch. 3.5 - Show that the inequality ( x2 ) 2 0 has one real...Ch. 3.5 - Explain why the inequality x 2 +x+10 has all real...Ch. 3.5 - Explain why the inequality x 2 x+10 has the empty...Ch. 3.5 - Explain the circumstances under which the...Ch. 3 - In Problems 1-3: (a) Determine the slope and...Ch. 3 - In Problems 1-3: (a) Determine the slope and...Ch. 3 - In Problems 1-3: (a) Determine the slope and...Ch. 3 - In Problems 4 and 5, determine whether the...Ch. 3 - In Problems 4 and 5, determine whether the...Ch. 3 - In Problems 6-8, graph each quadratic function...Ch. 3 - In Problems 6-8, graph each quadratic function...Ch. 3 - In Problems 6-8, graph each quadratic function...Ch. 3 - In Problems 9-14, (a) graph each quadratic...Ch. 3 - In Problems 9-14, (a) graph each quadratic...Ch. 3 - In Problems 9-14, (a) graph each quadratic...Ch. 3 - In Problems 9-14, (a) graph each quadratic...Ch. 3 - In Problems 9-14, (a) graph each quadratic...Ch. 3 - In Problems 9-14, (a) graph each quadratic...Ch. 3 - In Problems 15-17, determine whether the given...Ch. 3 - In Problems 15-17, determine whether the given...Ch. 3 - In Problems 15-17, determine whether the given...Ch. 3 - In Problems 18-19, solve each quadratic...Ch. 3 - In Problems 18-19, solve each quadratic...Ch. 3 - 20. In Problems 20 and 21, find the quadratic...Ch. 3 - 21. In Problems 20 and 21, find the quadratic...Ch. 3 - 22. Sales Commissions Bill has just been offered a...Ch. 3 - 23. Demand Equation the price p (in dollars) and...Ch. 3 - 24. Enclosing the Most Area with a Fence A farmer...Ch. 3 - 25. Minimizing Marginal Cost Callaway Golf Company...Ch. 3 - 26. Maximizing Area A rectangle has one vertex on...Ch. 3 - 27. Parabolic Arch Bridge A horizontal bridge is...Ch. 3 - 28. Bono Length Research performed at NASA, led by...Ch. 3 - 29. Advertising A small manufacturing firm...Ch. 3 - For the linear function f( x )=4x+3 , a. Find the...Ch. 3 - Determine whether the given function is linear or...Ch. 3 - Graph f(x)= (x3) 2 2 using transformations.Ch. 3 - In Problems 4 and 5, a. Determine whether the...Ch. 3 - In Problems 4 and 5, a. Determine whether the...Ch. 3 - Determine the quadratic function for the given...Ch. 3 - Determine whether f( x )=-2 x 2 +12x+3 has a...Ch. 3 - Solve, x 2 10x+240 .Ch. 3 - The weekly rental cost of a 20-foot recreational...Ch. 3 - The price p (in dollars) and the quantity x sold...Ch. 3 - Consider these two data sets: One data set follows...Ch. 3 - Find the distance between the points P=( 1,3 ) and...Ch. 3 - Which of the following points are on the graph of,...Ch. 3 - Solve the inequality 5x+30 and graph the solution...Ch. 3 - Find the equation of the line containing the...Ch. 3 - Find the equation of the line perpendicular to the...Ch. 3 - Graph the equation x 2 + y 2 4x+8y5=0 .Ch. 3 - Does the following relation represent a function?...Ch. 3 - For the function f defined by f( x )= x 2 4x+1 ,...Ch. 3 - Find the domain of h(z)= 3z1 6z7 .Ch. 3 - Is the following graph the graph of a function?Ch. 3 - Consider the function f(x)= x x+4 . a. Is the...Ch. 3 - Is the function f(x)= x 2 2x+1 even, odd, or...Ch. 3 - Approximate the local maximum values and local...Ch. 3 - If f(x)=3x+5 and g(x)=2x+1 , a. Solve f(x)=g( x )...Ch. 3 - For the graph of the function f , a. Find the...
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