find the dimensions of the largest area rectangle whose perimeter is 3600 feet. (enter the dimensions from smalles to largest) side: side:

Expert Answers

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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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