Question

A recatangular pen is to be made with 120 feet fencing. The pen is divided into 4 equal parts. And an existing property fence will be used for one long side.

1) If x represents the width of the fence express its area A(x) in terms of x

2) Determine the dimensions of the rectangle that will make area maximum

Answer 1

In my previous solution, I missed the part about needing only one long side.
So it is actually easier with no fractions.

long side ---- y ft
short side --- x ft each
y + 5x = 120
y = 120-5x

A(x) = xy = x(120-5x) = 120x - 5x^2
A'(x) = 120 - 10x
= 0 for a max of A(x)

10x = 120
x = 12 , same as before, but
y = 120-60 = 60

state the required conclusions

Answer 2

as always in this type of question, maximum area is achieved when the fence is divided equally among lengths and widths.
So, with only one length, that means it is 120/2 = 60.
The other 60 is divided into two widths, of 30 each.

The dimensions are this 60x30 for a maximum area of 1800.