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Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203

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Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203
Textbook Problem

Marginal Revenue and Profit 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 ) = 350 + 0.10 x + 0.002 x 2  dollars .

a. Calculate the marginal revenue and profit functions. [HINT: See Example 2.]

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

c. For which value of x is the marginal profit zero? Interpret your answer.

(a)

To determine

To calculate: The marginal revenue and profit function of annual fund-raising "Eatathon" by Audubon Society at ESU such that cost function of x servings of pasta is given by the function C(x)=350+0.10x+0.002x2 dollars where selling price of one serving is $1.10.

Explanation

Given Information:

The cost function of x servings of pasta is given by the function C(x)=350+0.10x+0.002x2 dollars where the selling price of one serving is $1.10.

Formula used:

The derivative of revenue R and profit P is marginal revenue and marginal profit.

A derivative of a function y=xn using power rule is dydx=nxn1.

A derivative of a constant is 0.

Constant multiple rules of derivative of a function f(x) are ddx[cf(x)]=cddx[f(x)] where c is constant.

Sum and difference rule of the derivative is ddx[f(x)±g(x)]=ddx[f(x)]±ddx[g(x)], there f(x) and g(x) are any two differentiable functions.

Calculation:

Consider the function,

C(x)=350+0.10x+0.002x2

The cost of one serving is $1.10 and there are x servings of pasta.

So,

Revenue function is,

R(x)=1.1x

Calculate the marginal revenue,

R(x)=ddx(1.1x)

Apply constant multiple and power rule of derivative,

R(x)=1

(b)

To determine

To calculate: The revenue, profit, marginal revenue and marginal profit if 200 servings of pasta are sold such that cost function of x servings of pasta is given by the function C(x)=350+0.10x+0.002x2 dollars where selling price of one serving is $1.10 and also interpret the answers.

(c)

To determine

To calculate: The value of x when marginal profit is 0 such cost function of x servings of pasta is given by the function C(x)=350+0.10x+0.002x2 dollars where selling price of one serving is $1.10 and also interpret the answers.

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