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
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
Answers
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
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
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.
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.
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