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# Step-by-Step Math Answer

Assign variables to all unknowns.

Let the length and width of the rectangle be
*
x
*
and
*
y
*
, respectively, and let the perimeter be
*
P
*
.

Find the primary equation.

The primary equation is
*
P
*
= 2
*
x
*
+ 2
*
y
*
.

Find the secondary equation.

The secondary equation is
*
xy
*
= 64.

Solve for
*
y
*
in the secondary equation.

Solving for
*
y
*
in the secondary equation we get
*
y
*
= 64/
*
x
*
.

Express the primary equation as a function of
*
x
*
.

Expressing the primary equation in terms of
*
x
*
,

we have
*
P
*
= 2
*
x
*
+ 2(64/
*
x
*
) = 2
*
x
*
+ 128/
*
x
*
, where 0
*
x
*
.

Differentiate
*
P
*
.

Differentiating
*
P
*
, we find that
*
P
*
′ (
*
x
*
) = 2 – 128/
*
x
*
^{
2
}
.

Find the critical number.

Solving the equation
*
P
*
′ (
*
x
*
) = 0, we find that the solutions are
*
x
*
= ± 8. Since
*
x
*
can't be negative, the only critical number is
*
x
*
= 8.

What should we do next?

Find
*
y
*
.

Verify that
*
P
*
is minimized for this value of
*
x
*
.

Let's verify that
*
P
*
is minimized for this value of
*
x
*
. Begin by finding the second derivative. Differentiating again, we find that
*
P
*
″ (
*
x
*
) = 256/
*
x
*
^{
3
}
.

Apply the Second Derivative Test.

Since
*
P
*
″ (8)
0,
*
P
*
is minimized when
*
x
*
= 8.

Use the secondary equation to find
*
y
*
.

Using the secondary equation, we find that
*
y
*
= 64/8 = 8. Therefore,
*
P
*
is at a minimum when
*
x
*
=
*
y
*
= 8 feet.