Chapter 3.4, Problem 28E

a.

To determine

To draw: The graph of $r(x)$ tofind the values of $x$ which makes sense in

the problem.

Answer to Problem 28E

  

The value at $x\to \infty$ , make sense in the problem.

Explanation of Solution

Given information:In the question, weekly revenue in dollars from selling x custom official desk is given $r(x)=2000\left(1-\frac{1}{x+1}\right)$ , using this information we will answer the following questions.

Calculation:See the graph carefully, from $x=200tox\to \infty$ value of $r(x)$ is going to constant.so at $x\to \infty$ the value of $r(x)$ is $2000$ .and this is constant.so this make sense in the problem.

  

Thus, the value at $x\to \infty$ , make sense in the problem.

b.

To determine

To calculate: The marginal revenue when x desks is sold.

Answer to Problem 28E

The Marginal revenue $\frac{dr}{dx}=\frac{2000}{{(x+1)}^2}$.

Explanation of Solution

Given information:In the question, weekly revenue in dollars from selling x custom official desk is given $r(x)=2000\left(1-\frac{1}{x+1}\right)$ , using this information we will answer the following questions.

Calculation: For finding marginal revenue we will differentiate of $r(x)$.

  $\text{Marginal cost=}\frac{\text{change in cost}}{\text{change in quantity}}$

  $\text{Marginal cost=}\frac{dr(x)}{dx}$

  $\frac{dr(x)}{dx}=\frac{2000}{{(x+1)}^2}$

Thus, the Marginal revenue $\frac{dr}{dx}=\frac{2000}{{(x+1)}^2}$.

c.

To determine

To use: $r^{\text{'}}(x)$ to find increase in sales desk.

Answer to Problem 28E

Increase in desk is $\text{500 dollars}$.

Explanation of Solution

Given information:In the question, weekly revenue in dollars from selling x custom official desk is given $r(x)=2000\left(1-\frac{1}{x+1}\right)$ , using this information we will answer the following questions.

Calculation: For finding increase in revenue from $5$ desks a week to $6$ desks a week, we will put increased value means $x=1$ in $\frac{dr}{dx}=\frac{2000}{{(x+1)}^2}$.

  $\begin{array}{l}\frac{dr}{dx}=\frac{2000}{{(x+1)}^2}\\ \frac{dr}{dx}=\frac{2000}{4}=500\end{array}$

Hence, $\text{500 dollars}$ is increased, when desks from $5$ to $6$.

d.

To determine

To Find : $\underset{x\to \infty }{\lim }{r}^{\text{'}}(x)$.

Answer to Problem 28E

  $\underset{x\to \infty }{\lim }{r}^{\text{'}}(x)=0$.

Explanation of Solution

Given information:In the question, weekly revenue in dollars from selling x custom official desk is given $r(x)=2000\left(1-\frac{1}{x+1}\right)$ , using this information we will answer the following questions.

Calculation:

(d) For finding limit value of $r^{\text{'}}(x)$ we will use limit concept.

  $\underset{x\to \infty }{\lim }{r}^{\text{'}}(x)=\underset{x\to \infty }{\lim }\frac{2000}{{(x+1)}^2}=0$

Thus, $\underset{x\to \infty }{\lim }{r}^{\text{'}}(x)=0$.