022 Exam 1 Sample A, Problem 5

Find the marginal revenue and marginal profit at $x=4$, given the demand function

$p={\frac {200}{\sqrt {x}}}$

and the cost function

$C(x)=100+15x+3x^{2}.$

Should the firm produce one more item under these conditions? Justify your answer.

Foundations:
Recall that the demand function, $p(x)$, relates the price per unit $p$ to the number of units sold, $x$. Moreover, we have several important important functions:
$C(x)$, the total cost to produce $x$ units; $R(x)$, the total revenue (or gross receipts) from producing $x$ units; $P(x)$, the total profit from producing $x$ units.
In particular, we have the relations
$P(x)=R(x)-C(x),$
and
$R(x)=x\cdot p(x).$
Finally, marginal profit at $x_{0}$ units is defined to be the effective cost of the next unit produced, and is precisely $P'(x_{0})$. Similarly, marginal revenue or cost would be $R'(x_{0})$ or $C'(x_{0})$, respectively.

 Solution:

Step 1:
Write the Basic Equation:

!Final Answer:   |- |With units, we have that the ladder is sliding down the wall at $-3/2$  feet per second. |}

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