   Chapter 13.4, Problem 35E ### 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

# The demand function for a certain product is p   = 144   — 2 x 2 and the supply function is p =   x 2 +   33 x   +   48 . Find the producer's surplus at the equilibrium point.

To determine

To calculate: The producer’s surplus at market equilibrium for a product whose demand function is approximated by p=1442x2 dollars and supply function is p=x2+33x+48 dollars.

Explanation

Given Information:

The producer’s surplus at market equilibrium for a product whose demand function is approximated by p=1442x2 dollars and supply function is p=x2+33x+48 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.

x=b±b24ac2a

Calculation:

Consider the supply function, p=x2+33x+48 and demand function, p=1442x2.

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

1442x2=x2+33x+483x2+33x96=03(x2+11x32)=0x2+11x32=0

Compare the above equation with the standard quadratic equation,

a=1b=11c=32

Use the quadratic formula to solve the quadratic equation to get the value of x,

x=b±b24ac2a

Substituting 1 for a, 11 for b and 32 for c to get:

x=11±(11)24(1)(32)2(1)=121±121+1282=2

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