find the dimensions of the largest rectangle whose perimeter is 4000 feet. (enter the dimensions from smallest to largest.) side: side:
Let us assume the dimensions as follws.
Length = xft
Width = yft
perimeter`(S) = x+x+y+y = 2(x+y)`
`2(x+y) = 4000`
` y = 2000-x`
Area of the rectangle` (A) = x*y`
`A = x(2000-x) = 2000x-x^2`
When the area is maximum or minimum then `(dA)/dx = 0`
`(dA)/dx = 2000-2x`
When `(dA)/dx = 0` ;
`2000-2x = 0`
`x = 1000`
If A is a maximum then `(d^2A)/(dx^2) < 0` at x = 1000
`(d^2A)/(dx^2) = -2 < 0`
`So (d^2A)/(dx^2) < 0`
This means A has a maximum.
The area will be maximum when;
x = 1000ft
y = 1000ft
At maximum area the rectangle will become a square.
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