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CalculusCalculus: An Applied Approach (MindTap Course List)10th Edition

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Chapter 5.5, Problem 57E
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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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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 function−price]dx

The producer surplus is defined as,

Producer surplus=∫0x0[price−supply function]dx

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

y−y1=(y2−y1x2−x1)(x−x1)

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,

p−331=(303−3318000−6000)(x−6000)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=Supply−0.014x+415=0.0275x−0.014x−0.0275x=0−415−0.0415x=−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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Getting Ready You should be able to complete these vocabulary and concept statements before you proceed to the ...

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Consider the following data. Period Rate of Return (%) 1 6.0 2 8.0 3 4.0 4 2.0 5 5.4 What is the mean growth ra...

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Exercises 3344 refer to lotteries; the order in which lottery numbers are chosen does not matter. A 6/49 lotter...

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