Chapter 9.9, Problem 35E

### Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781337625340

Chapter
Section

### Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781337625340
Textbook Problem

# The price of a product in a competitive market is $300. If the cost per unit of producing the product is 160 + 0.1 x dollars, where x is the number of units produced per month, how many units should the firm produce and sell to maximize its profit? To determine To calculate: The number of units of the product that the firm should produce and sell to maximize its profits. Explanation Given Information: The price of a product in the market is$300. The cost for producing one unit of the product is 160+0.1x dollars, where x represents the number of units produced per month.

Formula used:

The power rule of differentiation:

ddxxn=nxnâˆ’1

Calculation:

Consider the provided cost for 1 unit of the product, 160+0.1x. Thus, for x units of the product, the cost function is

C(x)=x(160+0.1x)=160x+0.1x2

The revenue earned from one unit of the product is \$300. Thus, for x units of the product, the revenue function is represented as,

R(x)=300x

The profit function is calculated by subtracting the revenue and cost functions,

P(x)=R(x)âˆ’C(x)=300xâˆ’160xâˆ’0.1x2=140xâˆ’0.1x2

To find the maximum profit, differentiate the profit function with respect to x,

Pâ€²(x)=ddx(140xâˆ’0.1x2)=ddx(140x)âˆ’ddx(0

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