find the dimensions of the largest area rectangle whose perimeter is 3600 feet. (enter the dimensions from smalles to largest) side: side:
Let us say length is X and width is Y.
Perimeter `= 2(X+Y) = 3600`
`2(X+Y) = 3600`
`X+Y = 1800`
`Y = 1800-X`
Area` (A) = X*Y = X(1800-X) = 1800X-X^2`
For maximum/minimum area dA/dX = 0
dA/dx = 1800-2X
When `(dA)/(dX) = 0` ;
`1800-2X = 0`
`X = 900`
If X = 900 has a maximum then `(d^2A)/(dX^2)` at X = 900 would be negative.
`(dA)/dx = 1800-2X`
`(d^2A)/dx^2 = -2 ` (negative)
So we have a maximum for the area.
X = 900
Y = 1800-900 = 900
So the dimensions would be a square with 900ft each side.
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