Question

A gift bag shaped like a rectangular prism has a volume of 1152 cubic inches. The dimensions of the gift bag in terms of its width are shown. The height is greater than the width. What are the dimensions of the gift bag? length: in., width: in., height: in.

Answer 1

The answer is
w = 6 in
l = 16 in
h = 12 in

The volume of a rectangular prism with length l, width w, and height h is:
V = l * w * h

V = 1152 in³

After some internet research, I found out that:
l = 2w + 4
h = 18 - w

So,
V = l * w * h
1152 = (2w + 4) * w * (18 - w)

Multiply the first two factors:
1152 = (2w² + 4w) * (18 - w)

Multiply two remaining factors:
1152 = 36w² + 72w - 2 w³ - 4w²

Rearrange:
-2w³ + 36w² - 4w² + 72w = 1152
-2w³ + 32w² + 72w = 1152
-2w³ + 32w² + 72w - 1152 = 0

Divide all by 2:
-w³ + 16w² + 36w - 576 = 0

Multiply by (-1):
w³ - 16w² - 36w + 576 = 0

Rearrange:
(w³ - 36w) - (16w² - 576) = 0

Factor:
w * w² - w * 36 - (16 * w² - 16 * 36) = 0
w(w² - 36) - 16(w² - 36) = 0
(w - 16)(w² - 36) = 0
(w - 16)(w² - 6²) = 0
(w - 16)(w - 6)(w + 6) = 0

So, w - 16 = 0, or w - 6 = 0, or w + 6 = 0.
In other words: w = 16, or w = 6, or w = -6.

Width cannot be negative, so w ≠ -6.
If w = 16, then l = 2 * 16 + 4 = 32 + 4 = 36 and h = 18 - 16 = 2
But, since the height must be greater than the width (h > w), w ≠ 16

If w = 6, then l = 2 * 6 + 4 = 12 + 4 = 16 and h = 18 - 6 = 12.
Thus:
w = 6 in
l = 16 in
h = 12 in