   Chapter 13.4, Problem 28E ### 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

# Suppose that the supply function for a good is p =   0.1 x 2 +   3 x +   20 . If the equilibrium price is \$36 per unit, what is the producer’s surplus there?

To determine

To calculate: The producer’s surplus for a good whose supply function is approximated by p=0.1x2+3x+20 dollars and the equilibrium price is 36 dollars per unit.

Explanation

Given Information:

The producer’s surplus for a good whose supply function is approximated by p=0.1x2+3x+20 dollars. Where, the equilibrium price is 36 dollars per unit.

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=0.1x2+3x+20.

The price of the unit at equilibrium is,

p=36

The standard form of the quadratic equation is,

ax2+bx+c

The provided supply function is given as,

p=0.1x2+3x+20

Substituting 36 for p in the supply function to get x1:

36=0.1x2+3x+20360=x2+30x+200x2+30x160=0

Compare the above equation with the standard quadratic equation,

a=1b=30c=160

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

x=b±b24ac2a

Substituting 1 for a, 30 for b and 160 for c to get:

x=30±(30)24(1)(160)2(1)=30±900+6402=4

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