# 022 Exam 1 Sample A, Problem 5

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Find the marginal revenue and marginal profit at ${\displaystyle x=4}$, given the demand function

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

and the cost function

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

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

Foundations:
Recall that the demand function, ${\displaystyle p(x)}$, relates the price per unit ${\displaystyle p}$ to the number of units sold, ${\displaystyle x}$.

Moreover, we have several important important functions:

• ${\displaystyle C(x)}$, the total cost to produce ${\displaystyle x}$ units;
• ${\displaystyle R(x)}$, the total revenue (or gross receipts) from producing ${\displaystyle x}$ units;
• ${\displaystyle P(x)}$, the total profit from producing ${\displaystyle x}$ units.
In particular, we have the relations
${\displaystyle P(x)=R(x)-C(x),}$
and
${\displaystyle R(x)=x\cdot p(x).}$
Finally, marginal profit at ${\displaystyle x_{0}}$ units is defined to be the effective cost of the next unit produced, and is precisely ${\displaystyle P'(x_{0})}$. Similarly, marginal revenue or cost would be ${\displaystyle R'(x_{0})}$ or ${\displaystyle C'(x_{0})}$, respectively.

Solution:

Step 1:
Write the Basic Equation:

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