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/ |
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