The width of a rectangular prism is w centimeters. The height is 2 centimers 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 dimensioon of the prism.

V = Lwh

w = width
w - 2 = height
w + 4 = length
8(w + 4) = V

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

Solve for w

w=2

To solve this problem, we need to use the given information to set up and solve equations. Let's go step by step:

Let's assume the width of the rectangular prism is represented by the variable w.

According to the problem, the height is 2 centimeters less than the width, so the height would be (w - 2) centimeters.

The length is 4 centimeters more than the width, so the length would be (w + 4) centimeters.

The volume of the prism is 8 times the measure of the length. Therefore, the volume is 8 times (w + 4).

To find the dimensions of the prism, we need to solve for w. So let's set up the equation:

Volume = Length × Width × Height
8(w + 4) = w × (w - 2) × (w + 4)

Now we can solve this equation to find the value of w:

8w + 32 = w^3 + 2w^2 - 4w

Rearranging the equation:

w^3 + 2w^2 - 4w - 8w - 32 = 0

Combining like terms:

w^3 + 2w^2 - 12w - 32 = 0

At this point, we can either use synthetic division, factoring, or use a tool like a graphing calculator to find the solution. After solving the equation, we find that one solution is w = 4.

Now that we know the width is 4 centimeters, we can find the height and length:

Height = w - 2 = 4 - 2 = 2 centimeters
Length = w + 4 = 4 + 4 = 8 centimeters

So the dimensions of the rectangular prism are:
Width = 4 centimeters
Height = 2 centimeters
Length = 8 centimeters