The Audubon Society at Enormous State University (ESU) is planning its annual fu

Question

The Audubon Society at Enormous State University (ESU) is planning its annual fund-raising "Eatathon." The society will charge students $1.10 per serving of pasta. The society estimates that the total cost of producing x servings of pasta at the event will be

C(x) = 310 + 0.10x + 0.002x2 dollars.

(a) Calculate the marginal revenue R'(x) and profit P'(x) functions.

R'(x)=___________

P'(x)=___________

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

Revenue- $_____

Profit- $____

Marginal Revenue- $____ per additional plate

Marginal Profit- $____ per additional plate

Interpret the results.

The approximate (profit of loss?) from the sale of the 201st 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.

Explanation / Answer

The society will charge students $1.10 per serving of pasta

So, For x pastas, revenue made = 1.1x dollars

So, revenue, R(x) = 1.1x

a) R'(x) = d/dx(1.1x) = 1.1 dollars per serving of pasta ---> ANSWER

P(x) = R(x) - C(x) = 1.1x - (310 + 0.10x + 0.002x^2)

P(x) = -0.002x^2 + x - 310

Deriving :

P'(x) = -0.004x + 1 ----> ANSWER

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b) 200 servings :

R(x) = 1.1x = 1.1*200 = $220 ---> ANSWER

P(x) = -0.002x^2 + x - 310 = -0.002(200)^2 + 200 - 310 = -$190.
So, a loss of 190 dollars ---> ANSWER

Marginal revenue = $1.1 per additional plate as found above ---> ANSWER

We have P'(x) = -0.004x + 1

Marginal profit , P'(200) = -0.004(200) + 1 = 0.2 dollars per additional plate --> ANSWER

The approximate (profit of loss?) from the sale of the 201st plate of pasta is $ :

P(201) = P(200) + P'(200)*(201 - 200)

P(201) = -190 + 0.2*1

P(201) = -189.80

So, the approximate LOSS from the sale of the 201st plate of pasta is $ 189.80 ---> ANSWER

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

P'(x) = -0.004x + 1 = 0

0.004x = 1

x = 1/0.004 = 250

So, for x = 250 ----> ANSWER

Graph :

P(x) = -0.002x^2 + x - 310

Here a = -0.002 , b = 1 and c = -310

Vertex x-value = -b/(2a) = -1 / (2*-0.002) = 250

When x = 250 , P(250) = -0.002(250)^2 + 250 - 310 = -185

So, answer is :

The graph of the profit function is a parabola with a vertex at x = 250 , so the loss is at a minimum when you produce and sell 250 plates.

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