SOLUTION: You have 180 yards of fencing to use on three sides of a garden. The fourth side is the tool shed. a. What is the maximum area that you can fence? b. What are the dimensions of

Algebra ->  Customizable Word Problem Solvers  -> Geometry -> SOLUTION: You have 180 yards of fencing to use on three sides of a garden. The fourth side is the tool shed. a. What is the maximum area that you can fence? b. What are the dimensions of      Log On

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Question 869209: You have 180 yards of fencing to use on three sides of a garden. The fourth side is the tool shed.
a. What is the maximum area that you can fence?
b. What are the dimensions of the garden?
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Let the dimensions of the area be x-y-x where y is the length parallel to the tool shed.
The area of the rectangle is A=xy
The amount of fencing used is L=2x%2By=148
Substitute into A,
A=x%28148-2x%29=148x-2x%5E2
To maximize A, find the derivative of area with respect to x.
dA%2Fdx=148-4x=0
4x=148
x=37
Then
y=148-2%2837%29
y=148-74
y=74
So,
a. A%5Bmax%5D=37%2A74=2738
b.x=37, y=74
.
.
.
If you had made a semi-circle with the fencing, then
pi%2AR=148
R=148%2Fpi
and the area would have been
A=%281%2F2%29pi%2AR%5E2}
A=%281%2F2%29pi%2A%28148%5E2%2Fpi%5E2%29
A=10952%2Fpi
A=3486 a 27% increase in garden area using the same amount of fence.