plywood box has a volume of 72 ft3. Its length is 9 ft greater than its height, and its width is 1 ft less than its height. What are the dimensions of the box?

Question

there should be 3 answers length, height, and width

Answer 1

Hi Michelle.

 

Here we have three variables: Length(L), Width(W) and Height(H)

 

Since the volume is 72 ft³,

LWH = 72

 

From the 2nd sentence

L = H+9

W = H-1

 

Since the length and width can be defined in terms of height, let's substitute these expressions for L and W into the first equation

 

LWH=72

(H+9)(H-1)H = 72

 

Let's foil this and multiply by H

H³ + 8H² - 9H = 72

H³ + 8H² - 9H -72 = 0

 

To factor this cubic, we can group the terms like this

(H³ + 8H²) - (9H +72) = 0

Factor each of the sets of parentheses

H²(H+8) - 9(H+8) = 0

Leaving us with

(H²-9)(H+8) = 0

 

We set these two factors equal to 0 to solver for H

H = -8, +3 -3

We can ignore the negative solutions because a measurement cannot be negative, so H=3

 

Use H=3 to solve for L and W

L = H+9

L = 3+9 = 12

W = H-1

W = 3-1 = 2

 

So the dimensions are

12x2x3