Find the length and width of a rectangle that has the given perimeter and a maximum area. (Enter your answers as a
comma-separated list.
Perimeter: units
units

Question

Find the length and width of a rectangle that has the given perimeter and a maximum area. (Enter your answers as a comma-separated list. Perimeter: $$ 5 P $$ units units

UpStudy AI · Accepted Answer

Let the rectangle have length $$ L $$ and width $$ W $$. The perimeter is given by

$$
2L + 2W = 5P.
$$

Dividing both sides by 2, we have

$$
L + W = \frac{5P}{2}.
$$

To maximize the area, we recall that for a fixed perimeter the rectangle with maximum area is a square. This means

$$
L = W.
$$

Substituting $$ W = L $$ into the perimeter equation,

$$
L + L = \frac{5P}{2},
$$

which simplifies to

$$
2L = \frac{5P}{2}.
$$

Solving for $$ L $$,

$$
L = \frac{5P}{4}.
$$

Since $$ L = W $$, we also have

$$
W = \frac{5P}{4}.
$$

Thus, the rectangle with maximum area has dimensions

$$
\frac{5P}{4}, \frac{5P}{4}.
$$
The rectangle has length and width of $$ \frac{5P}{4} $$ units each.

Find the length and width of a rectangle that has the given perimeter and a maximum area. (Enter your answers as a
comma-separated list.
Perimeter: units
units

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Find the length and width of a rectangle that has the given perimeter and a maximum area. (Enter your answers as a comma-separated list. Perimeter: units units

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Find the length and width of a rectangle that has the given perimeter and a maximum area. (Enter your answers as a
comma-separated list.
Perimeter: units
units