   # Cost In a manufacturing process where y = C ( x ) represents the cost of producing x units, the elasticity of cost is defined as E ( x ) = marginal cost average cost = C ′ ( x ) C ( x ) / x = x y d y d x . Find the cost function when the elasticity function is E ( x ) = 20 x − y 2 y − 10 x where C ( 100 ) = 500 and x ≥ 100. ### Multivariable Calculus

11th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781337275378

#### Solutions

Chapter
Section ### Multivariable Calculus

11th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781337275378
Chapter 16.1, Problem 45E
Textbook Problem
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## CostIn a manufacturing process where y = C ( x ) represents the cost of producing x units, the elasticity of cost is defined as E ( x ) = marginal   cost average   cost = C ′ ( x ) C ( x ) / x = x y d y d x .Find the cost function when the elasticity function is E ( x ) = 20 x − y 2 y − 10 x where C ( 100 ) = 500 and x ≥ 100.

To determine

The cost function when elasticity function is E(x)=20xy2y10x where C(100)=500 and x100.

### Explanation of Solution

Given information:

In a manufacturing process where y=C(x) represents the cost of producing x units, the elasticity of cost is defined as

E(x)=MarginalcostAveragecost=C(x)C(x)/x=xdyydx

Concept Used:

1) Let M and N have continuous partial derivatives on an open disk R. The differential equation

M(x,y)dx+N(x,y)dy=0

is exact if and only if

My=Nx

2) The equation M(x,y)dx+N(x,y)dy=0 is an exact differential equation when there exists a function f of two variables x and y having continuous partial derivatives such that

fx(x,y)=M(x,y) and fy(x,y)=N(x,y)

The general solution of the equation is fy(x,y)=C

Calculation:

By equating elasticity function we will get a differential equation:-

E(x)=xdyydx=20xy2y10x(20xy)ydx=(2y10x)xdy(20xyy2)dx+(10x22xy)dy=0

We can compare differential equation with the M(x,y)dx+N(x,y)dy=0.

M=20xyy2N=10x22xyMy=(20xyy2)yMy=20x2yNx=(10x22xy)xNx=20x2y

Since My=Nx, hence this differential equation is exact

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