Chapter 5.5, Problem 57E

### 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
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# Consumer and Producer Surpluses Factory orders for an air conditioner are about 6000 units per week when the price is $331 and about 8000 units per week when the price is$303. The supply function is given by p   = 0.0275 x . Find the consumer and producer surpluses. (Assume the demand function is linear.)

To determine

The consumer and producer surpluses of a factory for an air conditioner if the supply function is p=0.0275x.

Explanation

Given Information:

Factory orders are about 6000 units per week at the price of $331 and about 8000 units per week at the price of$ 303. The supply function is p=0.0275x, here, p is the price (in dollars) and x is the number of units per week.

Formula used:

The consumer surplus is defined as,

Consumer surplus=0x0[demand functionprice]dx

The producer surplus is defined as,

Producer surplus=0x0[pricesupply function]dx

The equation of a straight line passing through points (x1,y1) and (x2,y2) is,

yy1=(y2y1x2x1)(xx1)

Calculation:

Consider the supply functions,

p=0.0275x

Factory orders are about 6000 units per week at the price of $331 and about 8000 units per week at the price of$ 303. Take (x1,p1)=(6000, 331) and (x2,p2)=(8000, 303).

The linear function of demand is defined as,

p331=(30333180006000)(x6000)p=282000x+282000×6000+331p=0.014x+84+331p=0.014x+415

So, demand function is p=0.014x+415

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

Demand=Supply0.014x+415=0.0275x0.014x0.0275x=04150.0415x=415

Divide both sides by 0.0415,

x=(415)(0.0415)=10,000

Substitute x=10,000 in the demand function p=0.014x+415 and compute the price,

p0=0

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