   Chapter 11.3, Problem 58E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Production Suppose that production of 10,000 units of a certain agricultural crop is related to the number of hours of labor x and the number of acres of the crop y according to 300 x +   30 , 000 y   =   11 x y   − 0.0002 x 2 −   5 y Find the rate of change of the number of hours with respect to the number of acres.

To determine

To calculate: The rate of change of number of hours with respect to the number of acres if the number of hours of labor x and the number of acres of the crop y are related as 300x+30,000y=11xy0.0002x25y.

Explanation

Given Information:

The number of hours of labor x and the number of acres of the crop y are related as 300x+30,000y=11xy0.0002x25y.

Formula used:

According to the product rule of derivatives,

ddx(uv)=udvdx+vdudx

Calculation:

As it is provided that the number of hours of labor x and the number of acres of the crop y are related as:

300x+30,000y=11xy0.0002x25y.

Now, the rate of change of number of hours with respect to the number of acres is given by dxdy.

Now, differentiate both sides of equation with respect to y,

ddy(300x+30,000y)=ddy(11xy0.0002x25y)

To obtain the derivative of x2 as,

ddy(300x+30,000y)=ddy(11xy0.0002x25y)ddy(300x)+ddy(30,000y)=ddy(11xy)0

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