The width of a rectangular prism is w centimeters. The height is 2 centimeters less than the width. The length is 4 centimeters more than the width. If the volu

Question

The width of a rectangular prism is w centimeters. The height is 2 centimeters less than the width. The length is 4 centimeters more than the width. If the volume of the prism is 8 times the measure of the length, find the dimensions of the prism.

Answer 1

You know that volume is length times width times height.

 

First, define variables for all the dimensions.

 

width = w

height = w - 2

length = w + 4

Volume = 8(w + 4)

 

 

Using these variables, we can write the equation for volume.

 

width * height * length = volume

 

w(w - 2)(w + 4) = 8(w + 4)

 

 

Solve for w from this equation.

 

Dividing (w + 4) on both sides of the equation gives us

 

w(w - 2) = 8

 

 

Distribute the w to get rid of the parenthesis.

 

w² - 2w = 8

 

 

Subtract 8 on both sides of the equation to make the right side equal to zero.  This will give us a quadratic equation.

 

w² - 2w - 8 = 0

 

 

Using FOIL, we can factor the left side using two binomial factors.

 

(w - 4)(w + 2) = 0

 

 

w = 4            and          w = -2

 

 

Since we cannot have a negative value for dimensions, we accept the positive value of x:

 

x = 4

 

Your width is 4cm.

 

 

Substitute this value of x into the other dimensions to get your height and length.

 

height = 4 - 2

           = 2 cm

 

 

length = 4 + 4

          = 8  cm

 

 

Now we check our dimensions by multiply them to get the volume and then compare that volume to the length.  That quotient should give us 8.

 

 

Volume = 4 cm * 2 cm * 8 cm

            = 64 cm³

 

 

Volume / length = 64 / 8 = 8

 

 

This verifies the dimensions.