The length is three times the height and the height is one inch less than the width. The volume is 108 cubic inches.

Answer 1

You need to put this into equations form.

The length is three times the height can be written as L=3H

The height is one inch less than the width is written as H=W-1

And we know V=108.

The only thing left is to find an equation that includes all these variables: V=L*W*H for a rectangular prism.

We will put the equation in terms of W, so L=3H=3(W-1)=3W-3.

Now we plug in all the information into the volume equation: 108=(3W-3)*W*(W-1).

Multiply out the polynomial using the FOIL method:

108=(3W-3)*(W^2-W)

108=3W^3-3W^2-3W^2+3W divide by 3

36=W^3-W^2-W^2+W combine like terms

0=W^3-2W^2+W-36

When you get to this step you need to factor the polynomial using the p/q method. The possible factors are 1,2,4,6,9,18,and 36, all positive or negative. We can try plugging in each number to find whether it is a factor.

Lets try 1: (1)^3-2(1)^2+(1)-36 = -36 which is not 0, so 1 is not a factor.

Next, lets try 2: (2)^3-2(2)^2+(2)-36 = -34 which is also not a factor

Lets try 4: (4)^3-2(4)^2+(4)-36 = 0, so 4 is a factor.

We can now factor out (W-4) from the equation and are left with W^2+2W+9.

Since 4 is a factor we can check whether it fits the original equation for H, L, W, and V

H=4-1=3

L=3(3)=9

W=4

3*9*4=108 which is the total volume. Now we know we found the right answers for the side lengths.