A farmer wants to fence off a rectangular pen using part of an existing brick wall for one of the sides. He has 120 feet of fencing which he can use for this purpose and he wants to enclose as large an area as possible. What are the length and area of the pen of maximum area?

the area will be 900 ft, the perimeter will be 120 ft and the sides will be 30 ft each

Step-by-step explanation:

given that the area of the rectangular pen is

A = a*b

where "a" represents the first side (height) and "b" represents the second one (base). Then the total length of the fence (its perimeter) should be 120 feet:

2*a+2*b=120

b = (120 - 2*a) / 2 = 60-a

replacing b in A

A = a*b = a * (60-a) = 60*a - a²

if we complete the square

A = 60*a - a² - 30² + 30² = 30² - (a² - 60*a + 30²) = 900 ft - (a-30) ²

then A is minimum when a-30 = 0 → a=30 ft

thus

b=60 - a = 60 ft - 30 ft = 30 ft

thus the area will be

A = a*b = 30 ft * 30 ft = 900 ft (matches with our previous formula of A)

the perimeter will be 120 ft and the sides will be 30 ft each