Chapter 14.5, Problem 21E

### Mathematical Applications for the ...

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

Chapter
Section

### Mathematical Applications for the ...

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

# Cost A firm has two plants, X and Y. Suppose that the cost of producing x units at plant X  is  x 2 +   1200 dollars and the cost of producing y units of the same product at plant Y is given by 3 y 2 +   800 dollars. If the firm has an order for 1200 units, how many should it produce at each plant to fill this order and minimize the 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 has an order for 1200 units. It has two plants, X and Y and the cost of producing x units at plant X is x2+1200 dollars and the cost of producing y units of the same product at plant Y is given by 3y2+800 dollars.

Explanation

Given Information:

The cost of producing x units at plant X is x2+1200 dollars and the cost of producing y units of the same product at plant Y is given by 3y2+800 dollars and the firm has an order for 1200 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 Î» to form the objective function F(x,y,Î»)=f(x,y)+Î»g(x,y).

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

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

âˆ‚Fâˆ‚x=0, âˆ‚Fâˆ‚y=0, and âˆ‚Fâˆ‚Î»=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 fâ€²(x)=nxnâˆ’1.

Constant function rule for a constant c is such that, if f(x)=c then fâ€²(x)=0.

Coefficient rule for a constant c is such that, if f(x)=câ‹…u(x), where u(x) is a differentiable function of x, then fâ€²(x)=câ‹…uâ€²(x).

Calculation:

Consider the problem, the cost of producing x units at plant X is x2+1200 dollars and the cost of producing y units of the same product at plant Y is given by 3y2+800 dollars. The firm has an order for 1200 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.

Also, the firm has an order for 1200 units, thus x+y=1200.

So, the constraint is x+y=1200.

Also, if, the cost of producing x units at plant X is x2+1200 dollars and the cost of producing y units of the same product at plant Y is given by 3y2+800 dollars, then the total cost is

Cost=x2+1200+3y2+800.

Thus, minimize the cost function f(x,y)=x2+1200+3y2+800.

According to the Lagrange multipliers method,

The objective function is F(x,y,Î»)=f(x,y)+Î»g(x,y).

Here, f(x,y)=x2+1200+3y2+800 and g(x,y)=x+yâˆ’1200.

Substitute x2+1200+3y2+800 for f(x,y) and x+yâˆ’1200 for g(x,y) in F(x,y,Î»)=f(x,y)+Î»g(x,y)

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