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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 The demand and supply functions for a product are modeled by

D e m a n d : p = 60 / x 2 + 81 , S u p p l y : p = x / 3 where p is the price (in dollars) and x is the number of units (in millions). Find the consumer and producer surpluses for this product.

To determine

To calculate: The consumer and producer surpluses for the product, when the demand and the supply functions are modelled as p=60x2+81 and p=x3 respectively.

Explanation

Given Information:

The demand and the supply functions are modelled as p=60x2+81 and p=x3 respectively.

Formula used:

The formula for the integral 1u2±a2du is:

1u2±a2du=ln|u±u2±a2|+C

The General power of integration rule:

undu=un+1n+1+C

Here, n is the any number.

General power differentiation Rule:

ddx[un]=nun1dudx

Calculation:

Consider the provided demand and supply functions:

p=60x2+81

And,

p=x3

Now, Find the equilibrium point by equating functions of demand and supply.

60x2+81=x3(60)(3)=xx2+81180=xx2+81

Square both sides.

(xx2+81)2=(180)2x2(x2+81)=32400x4+81x232400=0x4+225x2144x232400=0

Further solve.

x2(x2+225)144(x2+225)=0(x2+225)(x2144)=0

Further solve.

x2144=0x2=144x2=122x=12

And,

x2+225=0x2=225x=225=15i(imaginary number)

Substitute 12 for x in expression p=x3 to calculate the price of product

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