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# Step-by-Step Math Answer

Assign variables to the unknowns.

Let
*
A
*
denote the area of the rectangle.

Find the length of the base of the rectangle.

From the graph, the length of the base of the rectangle is 2
*
x
*
.

Determine the height of the rectangle.

The height of the rectangle is
*
y
*
.

Find the primary equation.

The primary equation is
*
A
*
= 2
*
xy
*
.

Find the secondary equation.

The secondary equation is

Since the sides cannot be negative,
*
y
*
is defined for 25 –
*
x
*
^{
2
}
0.

Solve the inequality, 25 –
*
x
*
^{
2
}
0.

Solve the inequality, 25 –
*
x
*
^{
2
}
0.

(5 –
*
x
*
)(5 +
*
x
*
)
0

–(
*
x
*
– 5)(
*
x
*
+ 5)
0

(
*
x
*
– 5)(
*
x
*
+ 5)
0

This inequality is true for 0
*
x
*
5.

So,
*
y
*
is defined for 0
*
x
*
5.

Express the primary equation in terms of
*
x
*
.

In terms of
*
x
*
, we can express the primary equation as

Differentiate
*
A
*
.

Differentiating
*
A
*
, we find that

Find the critical numbers.

Solving the equation
*
A
*
′ (
*
x
*
) = 0, we find that the solutions are

However, negative values of
*
x
*
aren't allowed, so the critical number is

What should we do next?

Find the dimensions of the rectangle?

Verify that
*
A
*
is maximized for this value of
*
x
*
.

Let's verify that
*
A
*
is maximized for this value of
*
h
*
. Since
*
A
*
′ (3.5) = 0.28 and
*
A
*
′ (3.6) = –0.53, the First Derivative Test tells us that
*
A
*
is indeed maximized when

Find the dimensions of the rectangle.

The largest rectangle has a length of

The height of the largest rectangle is