How Does the Volume Change?

Date: 03/04/99 at 10:05:59
From: Caleb
Subject: How does the volume change when you change one side?

Recently in math class, we have been doing a project where we change 
one side of an object and see how it affects the volume. If you are 
doing this to a triangular prism and your formula is 1/2 the base 
times length times height, could you cut the 1/2 in half instead of 
the base, length, or height and get the same results?

Date: 03/04/99 at 13:06:10
From: Doctor Rick
Subject: Re: How does the volume change when you change one side?

It sounds as if you have noticed that whatever numbers you put in the 
formula, when you make the base, length or height half as big, the 
answer is the same as

  1/4 * base * length * height

and you want to know if this will always work.

If so, the answer is yes. What you are asking about is algebra. If you 
have a triangular prism and you reduce the length (for example) by a 
factor of 2, the volume of the new prism is

  Vnew = 1/2 base * (length/2) * height

You can rearrange the operations using the rules of algebra (the
commutative and associative principles, in particular). It does not 
matter what numbers base, length, and height stand for; these will 
always be equal:

1/2 * base * (length/2) * height

1/2 * base * (1/2 * length) * height     (commutative principle)

1/2 * (base * 1/2) * length * height     (associative principle)

1/2 * (1/2 * base) * length * height     (commutative principle again)

(1/2 * 1/2) * base * length * height     (associative principle again)

1/4 * base * length * height

This formula will work for any base, length, and height, because the
commutative and associative principles are true for any numbers.

If this is new to you, then you are discovering for yourself how 
algebra works, and that is great. If you have seen it before, I hope 
this helps you see how these principles can be put to work for you.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/