Question

question 9
1 pts
if $r(x)=50ln(4x + 1)$
and $c(x)=\frac{x}{4}$
find the marginal profit when $x = 200$ (round to the nearest cent)?

question 10
1 pts
based on your answer to the previous question, which answer is most true?
there will be little profit or loss when producing one more unit after the 200th unit
there is little cost to producing more than 200 items
profit is negligible when producing fewer than 200 items
the revenue after producing the 200th item is negligible

Explanation:

Step1: Recall profit formula

Profit $P(x)=R(x)-C(x)$. Given $R(x) = 50\ln(4x + 1)$ and $C(x)=\frac{x}{4}$.

Step2: Find marginal - profit formula

Marginal profit $P'(x)=R'(x)-C'(x)$. First, find $R'(x)$ and $C'(x)$.
For $R(x)=50\ln(4x + 1)$, using the chain - rule $\frac{d}{dx}\ln(u)=\frac{u'}{u}$ where $u = 4x+1$ and $u'=4$. So $R'(x)=\frac{50\times4}{4x + 1}=\frac{200}{4x + 1}$.
For $C(x)=\frac{x}{4}$, $C'(x)=\frac{1}{4}$.
Then $P'(x)=\frac{200}{4x + 1}-\frac{1}{4}$.

Step3: Evaluate marginal profit at $x = 200$

Substitute $x = 200$ into $P'(x)$:
\[

$$\begin{align*} P'(200)&=\frac{200}{4\times200+1}-\frac{1}{4}\\ &=\frac{200}{801}-\frac{1}{4}\\ &=\frac{200\times4-801}{801\times4}\\ &=\frac{800 - 801}{3204}\\ &=-\frac{1}{3204}\approx - 0.000312 \end{align*}$$

\]
Rounding to the nearest cent, $P'(200)\approx - 0.00$.

for Question 10:
The marginal profit at $x = 200$ is approximately $-0.00$. This means that there will be little profit or loss when producing one more unit after the 200th unit.

Answer:

• $0.00$