The Audubon Society at Enormous State University $($ESU$)$ is planning its annual fund$-$raising "Eatathon." The society will charge students

$$1\mathrm{.}10$ per serving of pasta. The society estimates that the total cost of producing $x$ servings of pasta at the event will be

$C(x)=330+0\mathrm{.}10x+0\mathrm{.}002x^2$ dollars.

$($a$)$ Calculate the marginal revenue $R^{\text{'}}(x)$ and profit $P^{\text{'}}(x)$ functions.

$R^{\text{'}}(x)=$

$P^{\text{'}}(x)=$

$($b$)$ Compute the revenue and profit, and also the marginal revenue and profit, if you have produced and sold $200$ servings of pasta $($in

$$.$

revenue $

profit $

marginal revenue $,$$$,$ per additional plate

marginal profit $$,$ per additional plate

Interpret the results.

The approximate

from the sale of the ${201}^{st}$ plate of pasta is $

$($c$)$ For which value of $x$ is the marginal profit zero?

$x=$

plates

Interpret your answer.

The graph of the profit function is a parabola with a vertex at $x=$

$,$ so the loss is at a minimum when you produce and

sell

plates.