Question

A rectangular playground is to be fenced off and divided in two by another fence parallel to one side of the playground. Six hundred feet of fencing is used. Find the dimensions of the playground that maximize the total enclosed area. What is the maximum area?

Answer 1

the smallest dimension = 100 ft

the largest dimension = 150 ft

Maximum Area = 15000 ft²

Explanation:

let x represent length of one side and L represent length of adjacent side

so from the question

3x + 2L = 600

2L = 600 - 3x

L = (600 - 3x)/2

L = (600/2) - (3x/2)

L = 300 - (3x/2)

now Area of rectangle = L × x

Area = (300 - (3x/2)) × x

Area = 300x - 3x²/2

now we differentiate with respect to x and equate to 0 in other to find critical point

dA/dx ⇒ 300x/x - 3x²/x = 0

300 - 3x = 0

3x = 300

x = 300/3

x = 100

so we input value of x into our previous equation

Area = 300x - 3x²/2

= 300(100) - (3(100)²)/2

= 30000 - 15000

Area = 15000

also we input value of x in L = 300 - (3x/2)

L = 300 - (3(100))/2)

L = 300 - 150

L = 150

Therefore

the smallest dimension = 100 ft

the largest dimension = 150 ft

Maximum Area = 15000 ft²