   Chapter 10, Problem 42RE ### 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

# Profit Suppose that for a product in a competitive market, the demand function is p   = 1200   − 2 x and the supply function is p =   200   + 2 x , where x is the number of units and p is in dollars. A firm’s average cost function for this product is C ¯ ( x ) = 12 , 000 x + 50 + x Find the maximum profit. (Hint: First find the equilibrium price.)

To determine

To calculate: The maximum profit if the demand function is p=12002x, and the supply function is p=200+2x for a product in a competitive market, where x is the number of units and p is in dollars.

Explanation

Given Information:

A firms average cost function for this product is C¯(x)=12000x+50+x.

Formula Used:

The total profit function is:

P(x)=R(x)C(x)

Where R(x)andC(x) are total revenue function and total cost function

Calculation:

Consider the provided equation,

C¯(x)=12000x+50+x

In case of equilibrium, it means that supply is equal to demand.

The cost at equilibrium can be calculated as follow:

Put the values of supply and demand equal:

12002x=200+2x

Solve it further for the value of x:

1200=200+4x

Deduct 200 from both the sides,

1000=4xx=10004x=250

Hence, the equilibrium price at x=250,

p=200+2(250)=200+500=700

The revenue as mentioned in the question is:

Revenue=700x, where x is the quantity.

The cost is:

C¯=12000x+50+xC=12,000+50x+x2

Hence, the profit function is equal to:

P=RC=700x(12,000+50x+x2)=700x12,00050xx2 …… (1)

Simplify the equation further,

P=700x12,00050xx2=12,000+650xx2

In order to maximize the profit, the following steps have to followed:

Take out the first derivative of the equation by the power rule,

Px

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