   Chapter 5.5, Problem 44E ### 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 = 300 − x                        p = 100 + x

To determine

To calculate: The consumer and producer surpluses for a product whose demand and supply functions are modeled by Demand:p=300x and Supply:p=100+x

Explanation

Given Information:

The demand and supply functions for a product are modeled by

Demand:p=300x and Supply:p=100+x

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=300x and Supply:p=100+x

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

Demand=Supply300x=100+xxx=1003002x=200

Divide both sides by 2,

(2x)(2)=(200)(2)x=100

Substitute x=100 in the demand function p=300x and compute the price,

p0=300100=200

So, the equilibrium point occurs at x=100 million and the price is \$ 200 per unit

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