Question

if total revenue received from the sale of x items is given by r(x)=10 ln(4x + 1), while the total cost to produce x items is c(x)=$\frac{x}{4}$, find the following
(a) the marginal revenue
(b) the profit function p(x)
(c) the marginal profit when x = 40
(d) interpret the results of part (c).

(a) how can the marginal revenue be found?
a. find the derivative of r(x).
b. find r(x)-c(x).
c. find r($\frac{x}{2}$).
d. find the derivative of r(x)-c(x).
the marginal revenue when x items are produced is .
(b) how can the profit function be found?
a. find r(x)-c(x)

Explanation:

Step1: Find marginal revenue

The marginal revenue is the derivative of the revenue function. Given $R(x)=10\ln(4x + 1)$. Using the chain - rule, if $y=\ln(u)$ and $u = 4x+1$, then $\frac{dy}{dx}=\frac{1}{u}\cdot\frac{du}{dx}$. Here, $\frac{dR}{dx}=\frac{10\cdot4}{4x + 1}=\frac{40}{4x+1}$.

Step2: Find profit function

The profit function $P(x)$ is given by $P(x)=R(x)-C(x)$. Since $R(x)=10\ln(4x + 1)$ and $C(x)=\frac{x}{4}$, then $P(x)=10\ln(4x + 1)-\frac{x}{4}$.

Step3: Find marginal profit

The marginal profit is the derivative of the profit function. $P^\prime(x)=\frac{40}{4x + 1}-\frac{1}{4}$.

Step4: Evaluate marginal profit at $x = 40$

Substitute $x = 40$ into $P^\prime(x)$. $P^\prime(40)=\frac{40}{4\times40+1}-\frac{1}{4}=\frac{40}{161}-\frac{1}{4}=\frac{40\times4-161}{161\times4}=\frac{160 - 161}{644}=-\frac{1}{644}$.

Step5: Interpret marginal profit

The marginal profit at $x = 40$ is $-\frac{1}{644}$. This means that when 40 items are produced and sold, the profit is decreasing at a rate of $\frac{1}{644}$ dollars per additional item produced and sold.

Answer:

(a) A. Find the derivative of $R(x)$, and the marginal revenue is $\frac{40}{4x + 1}$
(b) A. Find $R(x)-C(x)$, and the profit function is $P(x)=10\ln(4x + 1)-\frac{x}{4}$
(c) $-\frac{1}{644}$
(d) When 40 items are produced and sold, the profit is decreasing at a rate of $\frac{1}{644}$ dollars per additional item produced and sold.