   Chapter 11.3, Problem 57E ### 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 a company can produce 12,000 units when the number of hours of skilled labor y and unskilled labor x satisfy 384 = ( x + 1 ) 3 / 4 ( y + 2 ) 1 / 3 Find the rate of change of skilled-labor hours with respect to unskilled-labor hours when x = 255 and y = 214. This can be used to approximate the change in skilled-labor hours required to maintain the same production level when unskilled-labor hours are increased by 1 hour.

To determine

To calculate: The rate of change of skilled labor hours with respect to unskilled labor hours when the number of hours of skilled labor y and unskilled labor x satisfy 384=(x+1)3/4(y+2)1/3 at x=255 and y=214.

Explanation

Given Information:

The number of hours of skilled labor y and unskilled labor x satisfy, 384=(x+1)3/4(y+2)1/3.

Formula used:

According to the chain rule, if f and g are differentiable functions with y=f(u) and u=g(x), then y is a differentiable function of x,

dydx=dydududx

Calculation:

The number of hours of skilled labor y and unskilled labor x satisfy,

384=(x+1)3/4(y+2)1/3

Now, the rate of change of skilled labor hour with respect to unskilled labor hours is given by dydx.

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

ddx(384)=ddx((x+1)3/4(y+2)1/3)

To obtain the derivative of (x+1)3/4 and (y+2)1/3 as,

ddx(384)=ddx((x+1)3/4(y+2)1/3)0=(x+1)3/4ddx(y+2)1/3+(y+2)1/3ddx(x+1)3/40=(x+1)3/4(13)(y+2)

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