   Chapter 13.4, Problem 25E ### 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   =   1000   +   120 x   + 6 x 2 for its product, which has demand function p =   360   — 3 x   −   2 x 2 . 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=3603x2x2 and monopoly has a total cost function C=1000+120x+6x2.

Explanation

Given Information:

Demand function is approximated by p=3603x2x2 and

Monopoly has cost function C=1000+120x+6x2.

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=3603x2x2

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 to get,

R(x)=(3603x2x2)x=360x3x22x3

The Profit Function is,

P=RC

Substitute 360x3x22x3 for R and 1000+120x+6x2 for C to get profit function,

P=(360x3x22x3)(1000+120x+6x2)P=2x39x2+240x1000

Now differentiate the profit function to get the maxima,

p(x)=6x218x+240=6(x2+3x40)=6(x+8)(x5)

Now solve for x,</

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