Calculus question

Question

Determine the measurements x and y that will create a garden with a maximum area using 100 feet of fencing. Notice that a fence is only needed for three sides as the stone wall will serve as a boundary on the fourth side.

Answer 1

2x+y = 100

y=100-2x

Area = xy = x(100-2x) = 100x-2x^2

Take the derivative A'(x) = 100-4x set equal to zero and solve for x

100-4x = 0

4x = 100

x = 100/4 = 25

2x+y=100

2(25) + y = 100

50 + y = 100

y = 100 - 50 = 50

the side parallel to the wall is 50

the other sides are 25

It's a 25x50 foot area garden

maximum garden area with 100 feet of fencing is 25x50 = 1250 square feet