   Chapter 10.3, Problem 39E ### 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 monthly demand function for a product sold by a monopoly is p   = 1960  - 1 3 x 2 dollars, and the average cost is C ¯ = 1000 + 2 x + x 2 dollars. Production is limited to 1000 units, and x is in hundreds of units.(a) Find the quantity that will give maximum profit.(b) Find the maximum profit.

(a)

To determine

To calculate: The quantity that will give maximum profit if the monthly demand function for a product sold by a monopoly is p=196013x2 dollars, and the average cost is C¯=1000+2x+x2 dollars. Production is limited to 1000 units, and x is in hundreds of units.

Explanation

Given information:

The provided information is:

The monthly demand function for a product sold by a monopoly is p=196013x2 dollars, and the average cost is C¯=1000+2x+x2 dollars. Production is limited to 1000 units, and x is in hundreds of units.

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 information,

The monthly demand function for a product sold by a monopoly is p=196013x2 dollars, and the average cost is C¯=1000+2x+x2 dollars. Production is limited to 1000 units, and x is in hundreds of units.

The monthly demand function for a product sold by a monopoly is:

p=196013x2 dollars

Total cost function can be written as:

C(x)=C¯(x)x

Hence,

1000x+2x2+x3

Total revenue function can be written as:

R(x)=px

Hence, R(x)=1960x13x3

P(x) can be defined as:

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

(b)

To determine

To calculate: The maximum profit if the monthly demand function for a product sold by a monopoly is p=196013x2 dollars, and the average cost is C¯=1000+2x+x2 dollars. Production is limited to 1000 units, and x is in hundreds of units.

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