   Chapter 14.3, Problem 7E 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

The joint cost (in dollars) for two products is given by   C ( x ,   y ) =   30   +   x 2 + 3 y   +   2 x y where x represents the quantity of product X produced and y represents the quantity of product Y produced.(a) Find and interpret the marginal cost with respect to x if 8 units of product X and 10 units of product Y are produced.(b) Find and interpret the marginal cost with respect to y if 8 units of product X and 10 units of product Y are produced.

(a)

To determine

To calculate: The marginal cost with respect to x if 8 units of product X and 10 units of product Y are produced. The joint cost (in dollars) for two products is given by C(x,y)=30+x2+3y+2xy, where x represents the quantity of product X produced and y represents the quantity of product Y produced. Interpret the answer.

Explanation

Given Information:

The joint cost (in dollars) for two products is given by C(x,y)=30+x2+3y+2xy, where x represents the quantity of product X produced and y represents the quantity of product Y produced. Also, 8 units of product X and 10 units of product Y are produced.

Formula used:

For a joint cost function, of the form C(x,y), where x represents the quantity of product X and y represents the quantity of product Y, the marginal cost of the product X is given by Cx and the marginal cost of the product Y is given by Cy.

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. The partial derivative of f with respect to x is denoted by fx.

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

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 joint cost (in dollars) for two products is given by C(x,y)=30+x2+3y+2xy, where x represents the quantity of product X produced and y represents the quantity of product Y produced

(b)

To determine

To calculate: The marginal cost with respect to y if 8 units of product X and 10 units of product Y are produced. The joint cost (in dollars) for two products is given by C(x,y)=30+x2+3y+2xy, where x represents the quantity of product X produced and y represents the quantity of product Y produced. Also, interpret the answer.

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