Please help me figure out this answer.

Question

Suppose that a family wants to fence in an area of their yard for a vegetable garden to keep out deer. One side is already fenced from the neighbor's property. (a) If the family has enough money to buy 180 ft of fencing, what dimensions would produce the maximum area for the garden? (b) What is the maximum area?

Answer 1

180 feet of fencing would enclose a rectangular garden area with Area = A = LW

where L +2W = 180 or L = 180-2W

A = (180-2W)W = 180W - 2W^2

take the derivative and set equal to zero

A' = 180 -4W = 0

4W = 180

W = 180/4 = 45 feet wide

L = 180-2(45) = 90 feet long

90x45 = 4,050 square feet is the maximum area

dimensions are 90 x 45 feet

UNLESS you used a semi-circular garden

then use the fencing for a semi-circle. It would maximize the area more than a rectangle or any polygon shaped garden of any number of sides

circumference of a full circle = 2pi(r) = 180

r = 180/2pi = 90/pi

for a semi-circle r = 180/pi

Area = (pi(r)^2)/2 = pi(180/pi)^2 = 32,400/2pi = about 5,156.6 square feet = maximum area of the garden, with 180 meters of fencing, using one side without fencing. 5,156 > 4050 ft^2 for the rectangular garden