A rectangular picture of a purple cow is to be surrounded by a frame of uniform width/thickness

Question

A rectangular picture of a purple cow is to be surrounded by a frame of uniform width/thickness. The outside overall dimensions of the frame (including the picture) are 24 inches by 32 inches. Let w be the width/thickness of the frame.

a) Write an equation for area A of the picture in terms of the width of the frame.

Answer 1

A sketch will help you:

The area of the rectangular picture is its length multiplied by its width.

The length is "36 - 2w"

The width is "24 - 2w"

The area is (36 - 2w)(24 - 2w)

A(w) = (36 - 2w)(24 - 2w)

A(w) = [2(18 - w)][2(12 - w)]

A(w) = 4(18 - w)(12 - w)

A(w) = 4(216 - 18w - 12w + w²)

A(w) = 4(w² - 30w + 216)

A(w) = 4w² - 120w + 864