   Chapter 13.4, Problem 31E ### 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

# Find the producers surplus at market equilibrium for a product if its demand function is p =   81  -  x 2 and its supply function is p = x 2 + 4 x +   11 .

To determine

To calculate: The producer’s surplus at market equilibrium for a product whose demand function is approximated by p=81x2 dollars and supply function is p=x2+4x+11 dollars.

Explanation

Given Information:

The producer’s surplus at market equilibrium for a product whose demand function is approximated by p=81x2 dollars and supply function is p=x2+4x+11 dollars.

Formula used:

The producer’s surplus for a supply function g(x) at equilibrium is,

PS=p1x10x1g(x)dx.

Where p1 is the equilibrium price ad x1 is the unit sold at equilibrium.

Calculation:

Consider the supply function, p=4x2+2x+2 and demand function, p=81x2.

The supply function is equal to the demand function at equilibrium point.

81x2=x2+4x+112x2+4x70=02(x+7)(x5)=0

Further solve the above, to get

x=5

And

x=7 (Not possible, Since x can’t be negative)

Substituting x=5 in the demand function p=81x2 to get p1:

p1=81(5)2=8125=56

Thus, the equilibrium point (x1,p1)

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