Answer to Question #109517 in Calculus for Naledi

Question

A farmer with 750 m of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible area of the four pens? (Area of Rectangle =

Answer 1

Let the length of the sides of the rectangular with the largest possible area be x and y. Let us also assume that the fencing of the pens will be parallel to the side with the length y.

Then 2x + 5y = 750.

We are looking for the maximum of the function f(x,y) = xy with the restriction g(x,y) = 2x + 5y -750. We want to use the Lagrange multipliers method.

Let's consider the F(x,y) = f(x,y) + a*g(x,y). Since pair (x,y) is the one where f(x,y) has the maximum, we have the following:

(derivatives with respect to x and y are equal to 0 as well as g(x,y) = 0)

y= -2a

x= -5a

2x + 5y = 750

After solving this system we have a = -37.5. Thus, x = 187.5 and y = 75. This means that f(187.5, 75) = 14062.5 which is the answer (since we found only one extremum, it is the one we were looking for).