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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336
Textbook Problem

A company modeled the demand curve for its product (in dollars) by the equation

p = 800 , 000 e x / 5000 x + 20 , 000

Use a graph to estimate the sales level when the selling price is $16. Then find (approximately) the consumer surplus for this sales level.

To determine

To calculate: The sales level by using graph and the consumer surplus.

Explanation

Given information:

The equation of demand curve is p=800,000ex5,000x+20,000.

Selling price is p=$16.

Calculation:

Show the demand function as follows:

p(x)=800,000ex5,000x+20,000 (1)

Substitute 0 for x  in Equation (1).

p(0)=800,000e05,0000+20,000=800,00020,000=40

Similarly calculate the p(x) values for different x values as shown below.

x p(x)
0 40
600 34.44
1200 29.68
1800 25.60
2400 22.10
3000 19.09
3600 16.50
4200 14.27

Sketch the region as shown in Figure 1.

Refer to Figure 1.

The sales level is 3,726.83 for corresponding selling price as $16.

Therefore, the sales level is 3,726.83_ when the selling price is $16.

Find the producer surplus as shown below.

Consumersurplus=0x(p(x)p)dx (2)

Substitute 3,726.83 for x, 800,000ex5,000x+20,000 for p(x), and $16 for p in Equation (2).

Consumersurplus=03,726.83(800,000ex5,000x+20,00016)dx=1603,726.83(50,000ex5,000x+20,0001)dx=1603,726.8315,000(50,000ex5,000x5,000+4)dx1603,726.83dx (3)

Consider u=x5,000+4 (4)

Modify the Equation.

x=5,000(u4)

Differentiate both sides of the Equation

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