How Many Pyramids?Date: 08/21/2003 at 21:51:56 From: John Subject: Pyramids If a pyramid is 2 feet tall, and crumbles into sand, how many 2-inch pyramids can be created from the sand used to create the original pyramid? I figured it to be 650. (Based on height of 24", I assumed the base was 24" x 24", therefore there would be 12 x 12 = 144 for the first 2" row. Next would be 22" or 11x11 = 121, etc.) Date: 08/21/2003 at 22:41:18 From: Doctor Greenie Subject: Re: Pyramids Hi, John - If you can make a picture in your mind of your solution, you will see that there is a lot of empty space between the layers of pyramids, so there will be many more than 650 pyramids. This is a curious problem which is solved quite easily using an important geometric concept. That concept is the following: Suppose two figures are scale models of each other (mathematically, we say they are similar). Suppose they have a ratio of similarity of a:b, meaning that the ratio of any measure of length in the two figures is a:b. Then (1) the ratio of any measure of area between those two similar figures is a^2:b^2; and (2) the ratio of any measure of volume between those two similar figures is a^3:b^3 In your problem, the original pyramid has a height of 2 feet, or 24 inches; the small pyramids have a height of 2 inches. The ratio of those linear measurements is 24:2, or 12:1. With the ratio of similarity being 12:1, the ratio of the volumes of the two pyramids is 12^3:1^3, or 1728:1. That means the volume of the original pyramid is 1728 times the volume of each small pyramid; this in turn means you can make 1728 small pyramids from the sand of the original pyramid. I hope this helps. Please write back if you have any further questions about any of this. - Doctor Greenie, The Math Forum http://mathforum.org/dr.math/ |
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