A rectangular box has a perimeter of 36 inches. Find the length and width of the box that would give the maximum area.

e. l = 9, w = 9
f. l = 81, w = 9
g. l = 27, w = 9
h. l = 18, w = 18

length --- l

width ----w
2l+2w=36
l+w = 18
l = 18-w

area = lw = w(18-w) = 18w - w^2
d(area)/dw = 18-2w
= 0 for a max of area
2w = 18
w = 9
then l = 18-w = 9

so the area of the base of the box must be a square, 9 by 9
as you probably expected.

To find the length and width that would give the maximum area for a rectangular box with a perimeter of 36 inches, we can use the formula for the perimeter of a rectangle:

Perimeter = 2 * (length + width)

Given that the perimeter is 36 inches, we have:

36 = 2 * (length + width)

Simplifying the equation, we have:

18 = length + width

To find the maximum area, we need to maximize the product of the length and width. Since we have the equation length + width = 18, we can rewrite it as:

length = 18 - width

Substituting this expression for length in the formula for the area of a rectangle:

Area = length * width

we get:

Area = (18 - width) * width

To find the maximum area, we can find the value of width that gives the maximum value of Area.

Now let’s evaluate the options:

e. l = 9, w = 9:
Area = (18 - 9) * 9 = 9 * 9 = 81

f. l = 81, w = 9:
Area = (18 - 81) * 9 = -63 * 9 = -567 (Negative area is not possible.)

g. l = 27, w = 9:
Area = (18 - 27) * 9 = -9 * 9 = -81 (Negative area is not possible.)

h. l = 18, w = 18:
Area = (18 - 18) * 18 = 0 * 18 = 0 (Zero area is not possible.)

Based on the evaluation, the length and width that would give the maximum area is when:

Length = 9 and Width = 9

Therefore, the correct option is e. l = 9, w = 9.

To find the length and width of the box that would give the maximum area, we need to use the fact that the perimeter of a rectangle is given by the equation: P = 2L + 2W, where L is the length and W is the width.

In this case, we are given that the perimeter is 36 inches, so we can write the equation as: 36 = 2L + 2W.

To find the maximum area, we know that the area of a rectangle is given by the equation: A = L * W.

To maximize the area, we need to find the values of L and W that satisfy the given perimeter equation and result in the largest possible product L * W.

Let's check each option provided:

a. If we choose L = 9 and W = 9, the perimeter would be: 2 * 9 + 2 * 9 = 36 inches, which satisfies the given perimeter equation. The area would be: 9 * 9 = 81 square inches.

b. If we choose L = 81 and W = 9, the perimeter would be: 2 * 81 + 2 * 9 = 180 inches, which does not satisfy the given perimeter equation.

c. If we choose L = 27 and W = 9, the perimeter would be: 2 * 27 + 2 * 9 = 72 inches, which does not satisfy the given perimeter equation.

d. If we choose L = 18 and W = 18, the perimeter would be: 2 * 18 + 2 * 18 = 72 inches, which does not satisfy the given perimeter equation.

Therefore, the correct option is e. l = 9, w = 9, which gives us the maximum area of 81 square inches.