A rectangular play yard is to be constructed along the side of a house by erecting a fence on three sides, using house wall as the fourth wall. Find the demensions that produce the play yard of maximum area if 20 meters of fence is available for the project.

Answer 1

Make a sketch,
let the width of the field be x,
let the length of the field be y
then y + 2x = 20 , (we need only one length)
y = 20-2x

area = xy
= x(20-2x)
= -2x^2 + 20x

I don't know at what level of math-study you are.
If you know calculus, find
d(area)/dx = -4x + 20 = 0
x = 5 , then y = 10 for a max area of 50

If you don't know calculus, complete the square of the quadratic function
A = -2x^2 + 20x
= -2(x^2 - 10x + 25 - 25 )
= -2((x-5)^2 - 25)
= -2(x-5)^2 + 50

vertex is (5,50)
so the max area is 50 when x = 5 , as above