Chapter 7, Problem 102RE

### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

Chapter
Section

### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
1 views

# Average Revenue A company sells two products whose demand functions are given by x 1 = 500 − 2.5 p 1  and  x 2 =750-3 p 2 .The total revenue is given by R = x 1 p 1 + x 2 p 2 . Estimate the average revenue when price p 1 varies between $25 and$50 and price p 2 varies between $75 and$125.

To determine

To calculate: The average revenue of a company when the demand function of the two sold product is given by the function x1=5002.5p1 and x2=7503p2, and the total revenue is given by R=x1p1+x2p2.

Explanation

Given Information:

The demand function of the two sold product is given by the function,

x1=5002.5p1 and x2=7503p2

And the total revenue is given by;

R=x1p1+x2p2

Where price p1 varies between $25 and$50 and price p2 varies between $75 and$125.

Formula used:

When f is integrable over the plane region R with the area A, then the average value of the function f over the region R is given by;

Average value=1ARf(x,y)dA

The integration of function f(x)=xn,n1 is;

xndx=xn+1n+1+C

Calculation:

Consider the revenue function R=x1p1+x2p2.

Since price p1 varies between $25 and$50 and price p2 varies between $75 and$125

On plotting these vertices on rectangular coordinate system, a rectangle is formed with length 50 and width 25.

Thus, area of the rectangular region would be;

(50)(25)=1250

Since, x1=5002.5p1 and x2=7503p2 thus,

x1=5002.5p1p1=500x12.5=20025x1

And,

x2=7503p2p2=750x23=25013x2

Now substitute the value of p1 and p2 in the revenue function R=x1p1+x2p2 to obtain new revenue function as;

R=x1p1+x2p2=x1(20025x1)+x2(25013x2)=200x125x12+250x213x22

Here dx1 would be outer variable of integration with constant bounds 375x1437

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