Philip P. answered • 11/08/14

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Let L = the length of the rectangle and W = its width. The rectangle's perimeter is:

P = 192 = 2L + 2W

192 = 2L + 2W

192 - 2L = 2W

96 - L = W

The rectangle's area is:

A = L*W

Substitute 96-L in place of W:

A = L*(96-L)

A = -L

^{2}+ 96LThis is a quadratic equation whose graph is a parabola. Because the coefficient of the L

^{2}term is negative (-1), it's an inverted parabola with the vertex at the top. The vertex is thus the maximum value of the area A. The vertex is always located at the point**L = -b/2a**where**b**is the coefficient of the L term (96) and**a**is the coefficient of the L^{2}term (-1).L = -96/2(-1) = 48 inches

W = 96 - 48 = 48 inches.

**The rectangle of perimeter 192 inches that encloses the maximum area is a square 48 inches on a side.**----

You can also solve this using calculus. The area was:

A = -L

^{2}+ 96LTake the first derivative of A wrt L, set it to 0, and solve for L:

dA/dL = -2L + 96

0 = -2L + 96

2L = 96

L = 96/2 = 48 inches