I want to make a rectangular garden by fencing in a area on the back of my house. Three sides of the rectangle will be fencing; the fourth side will be the side of the house. I have 60 feet of fence. What dimensions should I make the rectangle so that I have the largest possible area for the garden?

Answer 1

W = The width of fence

L = Length of house = third side of fence

2 W + L = 60 ft

L = 60 - 2 W

A = W * L

A = W * ( 60 - 2 W )

A = 60 W - 2 W ^ 2

Now you must use calculus :

If a first derivative of a function at some point is equal to zero function has maxsimum or minimum.

If the second derivative is negative then the function has maximum.

If the second derivative f'' is positive then the function has minimum.

First derivative:

d A / d W = 60 - 2 * 2 W = 60 - 4 W

d A / d W = 60 - 4 W = 0

60 = 4 W Divide both sides by 4

15 = W

W = 15 ft

L = 60 - 2 W

L = 60 - 2 * 15

L = 60 - 30

L = 30 ft

Second derivative = - 4
the function has maximum.

Maximum area :

Amax = W * L = 15 * 30 = 450 ft ^ 2

Answer 2

thank you