Chapter 14.4, Problem 50AYU

Instantaneous Rate of Change The following data represent the total revenue $R$ (in dollars) received from selling $x$ bicycles at Tunney’s Bicycle Shop.

(a) Find the average rate of change in revenue from $x=25$ to $x=150$ bicycles.

(b) Find the average rate of change in revenue from $x=25$ to $x=102$ bicycles. per bicycle

(c) Find the average rate of change in revenue from $x=25$ to $x=60$ bicycles. per bicycle

(d) Using a graphing utility, find the quadratic function of best fit.

(e) Using the function found in part (d), determine the instantaneous rate of change of revenue at $x=25$ bicycles.

To determine

To find: Instantaneous Rate of Change The following data represent the total revenue $R$ (in dollars) received from selling $x$ bicycles at Tunney's Bicycle Shop.

a. Find the average rate of change in revenue from $x=25\text{to}x=150$ bicycles.

Answer to Problem 50AYU

Solution:

a. $249\frac{7}{25}$

Explanation of Solution

Given:

Calculation:

a. The average rate of change in revenue from $x=25tox=150$ is,

$\frac{\text{ΔR}}{\text{Δ<mi>x</mi>}}=\frac{R\left(150\right)-R\left(25\right)}{150-25}$

$R\left(150\right)=59,160;R\left(25\right)=28,000$

$\frac{\text{ΔR}}{\text{Δ}x}=\frac{R\left(150\right)-R\left(25\right)}{150-25}=\frac{59,160-28,000}{125}=\frac{31,160}{125}=249\frac{7}{25}$

To determine

To find: Instantaneous Rate of Change The following data represent the total revenue $R$ (in dollars) received from selling $x$ bicycles at Tunney's Bicycle Shop.

b. Find the average rate of change in revenue from $x=25\text{to}x=102$ bicycles. per bicycle.

Answer to Problem 50AYU

Solution:

b. $329\frac{67}{77}$

Explanation of Solution

Given:

Calculation:

b. The average rate of change in revenue from $x=25\text{to}x=102$ is,

$\frac{\text{ΔR}}{\text{Δ<mi>x</mi>}}=\frac{R\left(102\right)-R\left(25\right)}{102-25}$

$R\left(102\right)=53,400;R\left(25\right)=28,000$

$\frac{\text{ΔR}}{\text{Δ<mi>x</mi>}}=\frac{R\left(102\right)-R\left(25\right)}{102-25}=\frac{53,400-28,000}{77}=\frac{25,400}{77}=329\frac{67}{77}$

To determine

To find: Instantaneous Rate of Change The following data represent the total revenue $R$ (in dollars) received from selling $x$ bicycles at Tunney's Bicycle Shop.

c. Find the average rate of change in revenue from $x=25\text{to}x=60$ bicycles. per bicycle.

Answer to Problem 50AYU

Solution:

c. $\frac{17}{35}$

Explanation of Solution

Given:

Calculation:

c. The average rate of change in revenue from $x=25\text{to}x=60$ is,

$\frac{\text{ΔR}}{\text{Δ<mi>x</mi>}}=\frac{R\left(60\right)-R\left(25\right)}{60-25}$

$R\left(60\right)=45,000;R\left(25\right)=28,000$

$\frac{\text{ΔR}}{\text{Δ<mi>x</mi>}}=\frac{R\left(60\right)-R\left(25\right)}{60-25}=\frac{45,000-28,000}{35}=\frac{17}{35}$

To determine

To find: Instantaneous Rate of Change The following data represent the total revenue $R$ (in dollars) received from selling $x$ bicycles at Tunney's Bicycle Shop.

d. Using a graphing utility, find the quadratic function of best fit.

Answer to Problem 50AYU

Solution:

d. $-1.522x^2–597.791x+7944.45$

Explanation of Solution

Given:

Calculation:

d. Using a graphing utility, find the quadratic function of best fit.

The quadratic function is $y=-1.522x^2–597.791x+7944.45$.

To determine

To find: Instantaneous Rate of Change The following data represent the total revenue $R$ (in dollars) received from selling $x$ bicycles at Tunney's Bicycle Shop.

e. Using the function found in part $\left(d\right)$, determine the instantaneous rate of change of revenue at $x=25$ bicycles.

Answer to Problem 50AYU

Solution:

e. $-673.891$

Explanation of Solution

Given:

Calculation:

e. The instantaneous rate of change of $x_1$ is the derivative $R´\left(x_1\right)$: that is,

$R´\left(x_1\right)=\underset{x\to x_1}{\lim }\frac{R\left(x\right)-R\left(x_1\right)}{x-x_1}$

$=\underset{x\to x_1}{\lim }\frac{\left(-1.522x^2–597.791x+7944.45\right)-\left(-1.522x_1^2-597.791x_1+7944.45\right)}{t-t_1}$

$=\underset{x\to x_1}{\lim }\frac{-1.522\left(x^2-x_1^2\right)-597.791\left(x-x_1\right)}{x-x_1}$

$=\underset{x\to x_1}{\lim }\frac{-1.522\left[\left(x+x_1\right)\left(x-x_1\right)\right]-597.791\left(x-x_1\right)]}{x-x_1}$

$=\underset{x\to x_1}{\lim }\frac{\left(x-x_1\right)(-1.522\left(x+x_1)-597.791\right)}{x-x_1}$

$=\underset{x\to x_1}{\lim }(-1.522\left(x+x_1)-597.791\right)$

$=-1.522\left(x_1+x_1\right)–597.791$

Replace $x_1$ by $x$. the instantaneous rate of change is,

$R´\left(x\right)=-1.522\left(2x\right)–597.791$

The instantaneous rate of change at $x=25$ is,

$R´\left(25\right)=-1.522\left(2\left(25\right)\right)–597.791=-673.891$