   Chapter 10, Problem 13T ### 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 revenue function for a product is R ( x ) = 164 x dollars and the cost function for the product is C ( x )   =   0.01 x 2 + 20 x +   300 dollars where x is the number of units produced and sold.(a) How many units of the product should be sold to obtain maximum profit?(b) What is the maximum possible profit?

(a)

To determine

To calculate: The number of units of the product to be sold to obtain maximum profit where the revenue function is R(x)=164x and the cost function is C(x)=0.01x2+20x+300 where x is the number of units produced and sold.

Explanation

Given Information:

The provided revenue function is R(x)=164x and the cost function is C(x)=0.01x2+20x+300 where x is the number of units produced and sold.

Formula used:

To find relative maxima and minima of a function,

1. Set the first derivative of the function to zero, f'(x)=0, to find the critical values of the function.

2. Substitute the critical values into f(x) and calculate the critical points.

3. Calculate the sign of the function to the left and right of the critical values.

(a) If f(x)<0 and f(x)>0 on the left and right side respectively of a critical value, then the point is relative minimum.

(b) If f(x)>0 and f(x)<0 on the left and right side respectively of a critical value, then the point is relative maximum.

Calculation:

Consider the provided revenue function is R(x)=164x and the cost function is C(x)=0.01x2+20x+300 where x is the number of units produced and sold.

Since, profit is the difference of revenue and cost.

Thus, the profit function is the difference of revenue and cost function.

Let P represent profit.

Thus,

P(x)=164x(0.01x2+20x+300)=144x0.01x2300

Calculate the first derivative of the function P(x)=144x0.01x2300.

P(x)=1440

(b)

To determine

To calculate: The maximum profit where the revenue function is R(x)=164x and the cost function is C(x)=0.01x2+20x+300 where x is the number of units produced and sold.

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