   Chapter 14.3, Problem 19E 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

For Problems 17-19, suppose that the number of crates of an agricultural product is given by z = 11 x y − 0.0002 x 2 − 5 y 0.03 x + 3 y where x is the number of hours of labor and y is the number of acres of the crop.Find and interpret the marginal productivity of the number of hours of labor (x) when x = 300 and y = 500.

To determine

To calculate: The marginal productivity of the number of hours of labor (x) when x=300 and y=500 where the number of crates of an agricultural product is given by z=11xy0.0002x25y0.03x+3y where x is the number of hours of labor and y is the number of acres of the crop. Also, interpret the answer.

Explanation

Given Information:

The number of crates of an agricultural product is given by z=11xy0.0002x25y0.03x+3y where x is the number of hours of labor and y is the number of acres of the crop. Also, x=300 and y=500.

Formula used:

For a production function, of the form z=f(x,y), the marginal productivity of x is given by zx and the marginal productivity of y is given by zy.

For a function f(x,y), the partial derivative of f with respect to y is calculated by taking the derivative of f(x,y) with respect to y and keeping the other variable x constant. The partial derivative of f with respect to y is denoted by fy.

Power of x rule for a real number n is such that, if f(x)=xn then f(x)=nxn1.

Quotient rule for function f(x)=u(x)v(x), where u and v are differentiable functions of x, then f(x)=v(x)u(x)u(x)v(x)(v(x))2.

Constant function rule for a constant c is such that, if f(x)=c then f(x)=0.

The coefficient rule for a constant c is such that, if f(x)=cu(x), where u(x) is a differentiable function of x, then f(x)=cu(x).

Calculation:

Consider the problem,

The number of crates of an agricultural product is given by z=11xy0.0002x25y0.03x+3y where x is the number of hours of labor and y is the number of acres of the crop.

For a production function of the form z=f(x,y), the marginal productivity of x is given by zx.

Recall that, for a function f(x,y), the partial derivative of f with respect to x is calculated by taking the derivative of f(x,y) with respect to x and keeping the other variable y constant.

Use the power of x rule for derivatives, the quotient rule, the constant function rule, and the coefficient rule,

zx=(0.03x+3y)(11y0.0002(2x))(11xy0.0002x25y)(0.03)(0.03x+3y)2=(0

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