   Chapter 13.4, Problem 26E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# In Problems 17-26, p and C are in dollars and x is the number of units.A monopoly has a total cost function   C   =   500   + 2 x 2 + 10 x for its product, which has demand function p   = — 1 3 x 2 − 2 x   + 30 Find the consumers surplus at the point where the monopoly has maximum profit.

To determine

To calculate: The Consumer’s Surplus where the monopoly has maximum profit for a product whose demand function is approximated by p=13x22x+30 and monopoly has cost function C=500+2x2+10x.

Explanation

Given Information:

Demand function is approximated by p=13x22x+30 and

Monopoly has cost function C=500+2x2+10x.

Formula used:

According to the Consumer’s Surplus

If the demand curve has equation p=f(x), the consumer surplus is given by the area between f(x) and the x-axis from 0 to x1, less the area of TR.

CS=0x1f(x)dxp1x1.

The Profit Function is:

P=RC

Calculation:

Consider the demand function,

p=13x22x+30

Since, to know the point where the profit is maximized that is R(x),

R(x)=p(x)

Substitute the value of p in above equation,

R(x)=(13x22x+30)x=13x32x2+30x

Thus, the Profit Function is:

P=RC

Substitute 13x32x2+30x for R and 500+2x2+10x for C to get profit function,

P=(13x32x2+30x)(500+2x2+10x)=13x34x2+20x500

Differentiate the profit function to get the maxima,

p(x)=x28x20=(x+10)(x2)=0

Now solve for x,

x=2

And

x=10 (Not possible, since x can’t be negative)

Differentiate the profit function p(x) again,

p(x)=2x8

The value of this function at x=2

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