Question Video: Writing the Relation between Two Quantities for the Volume of Rectangular Prisms

Video Transcript

Which of the following describes how the volume of a rectangular prism is affected after doubling all three dimensions? Is it (A) 𝑉 sub new is equal to six times 𝑉 sub old? (B) 𝑉 new is equal to 𝑉 old squared. (C) 𝑉 new is equal to two times 𝑉 old. (D) 𝑉 new is equal to four times 𝑉 old. Or (E) 𝑉 new is equal to eight times 𝑉 old.

To answer this question, we’ll begin by recalling the formula for the volume of a rectangular prism. The volume 𝑉 of a rectangular prism is the product of its length, its width, and its height. So, let’s call the volume of our original shape 𝑉 old. It’s 𝑤𝑙ℎ. Now, of course, we could do this in any order. So, we could write 𝑙𝑤ℎ or any other combination. We’re now going to take our original rectangular prism and double all of its dimensions.

The height of our new shape is two times ℎ, which is two ℎ. The width is now two times 𝑤. That’s two 𝑤. And the length is two times 𝑙. That’s two 𝑙. And so, we can now calculate the volume of the new shape. It’s still the product of all of its dimensions, but this time that’s two 𝑤 times two 𝑙 times two ℎ. When we multiply algebraic expressions, such as this, we begin by multiplying the numbers. And so, two times two times two is eight. And the new volume is eight 𝑤𝑙ℎ.

We now compare the original volume to the new volume. And since the original volume is 𝑤𝑙ℎ and the new volume is eight times this, this must mean that the new volume is eight times the old volume. The correct answer, therefore, is (E) 𝑉 sub new is equal to eight times 𝑉 sub old.