   Chapter 5.5, Problem 45E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

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

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Consumer and Producer Surpluses In Exercises 43-48, find the consumer and producer surpluses by using the demand and supply functions, where p is the price (in dollars) and x is the number of units (in millions). See Example 5.Demand Function Supply Function p = 200 − 0.4 x                           p = 100 + 1.6 x

To determine

To calculate: The consumer and producer surpluses for a product whose demand and supply functions are modeled by Demand:p=2000.4x and Supply:p=100+1.6x

Explanation

Given Information:

The demand and supply functions for a product are modeled by

Demand:p=2000.4x and Supply:p=100+1.6x

Here, p is the price (in dollars) and x is the number of units (in millions).

Formula used:

The consumer surplus is the area of the region bounded by the graph of the demand function, the horizontal line p=p0 and the vertical line x=0.

Consumer surplus=0x0[demand functionprice]dx

The producer surplus is the area of the region bounded by the graph of the supply function, the horizontal line p=p0 and the vertical line x=0.

Producer surplus=0x0[pricesupply function]dx

Here (x0,p0) is the equilibrium point at which the demand function and supply function intersect.

Calculation:

Consider the demand and supply functions,

Demand:p=2000.4x and Supply:p=100+1.6x

Compute the points of intersection of two graphs by setting the functions equal to each other and solving for x,

Demand=Supply2000.4x=100+1.6x0.4x1.6x=1002002.0x=100

Divide both sides by 2.0,

(2.0x)(2.0)=(100)(2.0)x=50

Substitute x=50 in the demand function p=2000.4x and compute the price,

p0=2000.4×50=20020=180

So, the equilibrium point occurs at x=50 million and the price is \$ 180 per unit.

Use the formula for the consumer surplus and calculated it for the provided product as follows:

Consumer surplus=0x0[demand functionprice]dx=050[(2000

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