The aspect ratio of the envelope .

(a)

The aspect ratio of the envelope .

Formula for the area of a rectangle .

Area of envelope .

Substitute in .

The area of the envelope as a function of length is .

The minimum allowable length is inches and the maximum allowable length is inches.

The domain of the function is all real numbers that greater than or equal to the minimum allowable length of inches and less than or equal to the maximum allowable length of inches.

The domain is in interval notation .

(b)

The aspect ratio of the envelope .

Area of a rectangle .

Area of envelope .

Substitute  in .

The area of the envelope as a function of height is .

Find the minimum and maximum allowable heights for the envelope.

The minimum allowable length is inches.

Substitute .

Minimum allowable height is inches.

The maximum allowable length is inches.

Substitute in .

Maximum allowable height is inches.

The domain of the function is all real numbers that greater than or equal to the minimum allowable height of inches and less than or equal to the maximum allowable length of inches.

The domain is in interval notation .

(c)

Find the area of the envelope with the maximum height at the maximum aspect ratio:

The aspect ratio of the envelope is and the length is .

The aspect ratio of the envelope .

Substitute and in .

.

The area of the envelope with the maximum height at the maximum aspect ratio is square inches.

(a) The domain in interval notation is .

(b) The domain in interval notation is .

(c) The area of an envelope with the maximum height at the maximum aspect ratio is square inches.