   Chapter 14.5, Problem 22E 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

Cost Suppose that the cost of producing x units at plant X  is  ( 3 x   +   4 ) x dollars and that the cost of producing y units of the same product at plant Y  is ( 2 y   +   8 ) y dollars. If the firm that owns the plants has an order for 149 units, how many should it produce at each plant to fill this order and minimize its cost of production?

To determine

To calculate: The number of units produced at each plant to fill the order and minimize the cost of production if the firm that owns the plant has an order for 149 units. The cost of producing x units at plant X is (3x+4)x dollars and the cost of producing y units of the same product at plant Y is given by (2y+8)y dollars.

Explanation

Given Information:

The cost of producing x units at plant X is (3x+4)x dollars and the cost of producing y units of the same product at plant Y is given by (2y+8)y dollars. The firm has an order for 149 units.

Formula used:

Lagrange Multipliers Method:

According to the Lagrange multipliers method to obtain maxima or minima for a function z=f(x,y) subject to the constraint g(x,y)=0,

Step 1: Find the critical values of f(x,y) using the new variable λ to form the objective function F(x,y,λ)=f(x,y)+λg(x,y).

Step 2: The critical points of f(x,y) are the critical values of F(x,y,λ) which satisfies g(x,y)=0.

Step 3: The critical points of F(x,y,λ) are the points that satisfy:

Fx=0, Fy=0, and Fλ=0, that is, the points which make all the partial derivatives of zero.

For a function f(x,y), the partial derivative of f(x,y) 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(x,y) 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.

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

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 cost of producing x units at plant X is (3x+4)x dollars and the cost of producing y units of the same product at plant Y is given by (2y+8)y dollars. The firm has an order for 149 units.

If x units are produced at plant X and if y units are produced at plant Y, then total units produced is x+y. But the firm has an order for 149 units. Thus, x+y=149.

Thus, the constraint is x+y=149.

Also, if, the cost of producing x units at plant X is (3x+4)x dollars and the cost of producing y units of the same product at plant Y is given by (2y+8)y dollars, then the total cost is:

Cost=(3x+4)x+(2y+8)y=3x2+4x+2y2+8y dollars.

Thus, minimize the cost function f(x,y)=3x2+4x+2y2+8y.

According to the Lagrange multipliers method,

The objective function is F(x,y,λ)=f(x,y)+λg(x,y).

Here, f(x,y)=3x2+4x+2y2+8y and g(x,y)=x+y149.

Substitute 3x2+4x+2y2+8y for f(x,y) and x+y149 for g(x,y) in F(x,y,λ)=f(x,y)+λg(x,y)

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