   Chapter 5.5, Problem 46E ### 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 = 975 − 23 x                             p = 42 x

To determine

To calculate: The consumer and producer surpluses for a product whose demand and supply functions are modeled by Demand:p=97523x and Supply:p=42x

Explanation

Given Information:

The demand and supply functions for a product are modeled by

Demand:p=97523x and Supply:p=42x

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=97523x and Supply:p=42x

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

Demand=Supply97523x=42x23x42x=097565x=975

Divide both sides by 65,

(65x)(65)=(975)(65)x=15

Substitute x=15 in the demand function p=97523x and compute the price,

p0=97523×15=975345=630

So, the equilibrium point occurs at x=15 million and the price is \$ 630 per unit

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