Chapter 4.7, Problem 36PPS

a.

To determine

Write a function that represents the total monthly cost C(x) of M’s cable package , wherex is the number of additional movies watched.

Answer to Problem 36PPS

  $C(x)=5.50x+108.$

Explanation of Solution

Given information: M pays a monthly fee for her cable package which includes

one TV movie a month.M gets billed an additional charge for every additional movie.During her first month,M watched 5 additional movies and her bill was $135.50.During her second month.M watched 9 additional movies and her bill was $157.50.

Formula used:

The equation of a line in point-slope form is $y=mx+b.$ where m is the slope and b is the y -intercept of the line.

Slop m is calculated by $m=\frac{y_2-y_1}{x_2-x_1}$ .

Calculation:

If 5 Additional movies costs $157.50,then the function passes through the point(5,135.5).If 9 Additional movies costs $157.50,then the function passes through the point(9,155.50).So equation of the function passes through points (5,135.50) and (9,157.50) is,

  $\begin{array}{l}y-y_1=m(x-x_1).\\ m=(y_2-y_1)/(x_2-x_1).\\ m=(157.50-135.50)/9-5=22/4=\mathrm{5.50.}\\ y-135.50=5.50(x-5).\\ y=5.50x+108.\\ C(x)=5.50x+108.\end{array}$

b.

To determine

Find the inverse function.

Answer to Problem 36PPS

  $C^{-1}(x)=(2/11)x-(216/11).$

Explanation of Solution

Given information:M pays a monthly fee for her cable package which includes

one TV movie a month .M gets billed an additional charge for every additional movie .During her first month ,M watched 5 additional movies and her bill was $135.50.During her second month .M watched 9 additional movies and her bill was $157.50.

Formula used:

The equation of a line in point-slope form is $y=mx+b.$ where m is the slope and b is the y -intercept of the line.

Slop m is calculated by $m=\frac{y_2-y_1}{x_2-x_1}$ .

Calculation:

If 5 Additional movies costs $157.50,then the function passes through the point (5,135.5).If 9 Additional movies costs $157.50,then the function passes through the point (9,155.50).So equation of the function passes through points (5,135.50) and (9,157.50) is,

  $\begin{array}{l}y-y_1=m(x-x_1).\\ m=(y_2-y_1)/(x_2-x_1).\\ m=(157.50-135.50)/9-5=22/4=\mathrm{5.50.}\\ y-135.50=5.50(x-5).\\ y=5.50x+108.\\ C(x)=5.50x+108.\end{array}$

  $\begin{array}{l}y=5.50x+108.\\ \text{switching}x\text{and}y.\\ x=5.50y+108.\\ 5.50y=x-108.\\ y=(x/5.50)-(108/5.50).\\ C^{-1}(x)=(2/11)x-(216/11).\end{array}$

c.

To determine

To find x and $C^{-1}(x)$ represent in the context of the inverse function.

Answer to Problem 36PPS

xis the cost and $C^{-1}(x)$ is the number of additional movies.

Explanation of Solution

Given information:M pays a monthly fee for her cable package which includes

one TV movie a month .M gets billed an additional charge for every additional movie .During her first month ,M watched 5 additional movies and her bill was $135.50.During her second month .M watched 9 additional movies and her bill was $157.50.

Formula used:

The equation of a line in point-slope form is $y=mx+b.$ where m is the slope and b is the y -intercept of the line.

Slop m is calculated by $m=\frac{y_2-y_1}{x_2-x_1}$ .

Calculation:

If 5 Additional movies costs $157.50,then the function passes through the point (5,135.5).If 9 Additional movies costs $157.50,then the function passes through the point (9,155.50).So equation of the function passes through points (5,135.50) and (9,157.50) is,

  $\begin{array}{l}y-y_1=m(x-x_1).\\ m=(y_2-y_1)/(x_2-x_1).\\ m=(157.50-135.50)/9-5=22/4=\mathrm{5.50.}\\ y-135.50=5.50(x-5).\\ y=5.50x+108.\\ C(x)=5.50x+108.\end{array}$

  $\begin{array}{l}y=5.50x+108.\\ \text{switching}x\text{and}y.\\ x=5.50y+108.\\ 5.50y=x-108.\\ y=(x/5.50)-(108/5.50).\\ C^{-1}(x)=(2/11)x-(216/11).\end{array}$

In the original function x was the number of movies and $C(x)$ was cost .For the inverse function, x is then the cost and $C^{-1}(x)$ is the number of additional movies.

d.

To determine

To find how many additional movies did M watched if her bill for her third month was $174.

Answer to Problem 36PPS

12 Movies.

Explanation of Solution

Given information: M pays a monthly fee for her cable package which includes

one TV movie a month .M gets billed an additional charge for every additional movie .During her first month ,M watched 5 additional movies and her bill was $135.50.During her second month .M watched 9 additional movies and her bill was $157.50.

Formula used:

The equation of a line in point-slope form is $y=mx+b.$ where m is the slope and b is the y -intercept of the line.

Slop m is calculated by $m=\frac{y_2-y_1}{x_2-x_1}$ .

Calculation:

If 5 Additional movies costs $157.50,then the function passes through the point (5,135.5).If 9 Additional movies costs $157.50,then the function passes through the point (9,155.50).So equation of the function passes through points (5,135.50) and (9,157.50) is,

  $\begin{array}{l}y-y_1=m(x-x_1).\\ m=(y_2-y_1)/(x_2-x_1).\\ m=(157.50-135.50)/9-5=22/4=\mathrm{5.50.}\\ y-135.50=5.50(x-5).\\ y=5.50x+108.\\ C(x)=5.50x+108.\end{array}$

  $\begin{array}{l}y=5.50x+108.\\ \text{switching}x\text{and}y.\\ x=5.50y+108.\\ 5.50y=x-108.\\ y=(x/5.50)-(108/5.50).\\ C^{-1}(x)=(2/11)x-(216/11).\end{array}$

  $\begin{array}{l}{C}^{-1}(175)=(2/11)\times (175)-(216/11).\\ C^{-1}(175)=(348/11)-(216/11).\\ C^{-1}(175)=12.\end{array}$