Help me please The length, width and height of a small box are three consecutive integers, where the width

Question

The length, width and height of a small box are three consecutive integers, where the width is

the least and the length is the greatest integer. If the width is doubled, and the length and

height are increased by 2cm each, the volume is increased by 96cm3Find the width, height,and length of the original box.

Answer 1

Since the order of the dimensions is width, height, length and the numbers are consecutive integers, we know:

H = W + 1

L = H + 1 = W + 2

We also know:

2W * (2 + H) * (2 + L) = V + 96

L * W * H = V

We can solve for W and V using the equations above:

2W * (2 + W + 1) * (2 + W + 2) = V + 96

W * (W + 1) * (W + 2) = V

2W(W + 3)(W + 4) - 96 = W(W + 1)(W + 2)

2W(W^2 + 7W + 12) - 96 = W(W^2 + 3W +2)

2W^3 + 14W^2 + 24W - 96 = W^3 + 3W^2 + 2W

W^3 + 11W^2 + 22W - 96 = 0

Using a graphing calculator, W = 2 cm, H = 3 cm and L = 4 cm