Monday, June 23, 2008

Length, Area, and Volume

Hello, I'm Mama Joules' husband. Julie has asked me to write a blog post for her. I have loved science since I was a little kid, and still find all kinds of questions fascinating. I work as a researcher in cryptography, and I usually try to understand things in terms of math, even stuff like biology.

Have you ever wondered why ants can't be ten feet tall? The answer has to do with scaling. If you made an ant ten feet tall, all kinds of things inside the ant would stop working, because they wouldn't scale up well. The same thing applies in a lot of other places. Even if you made a perfect scale model of an airplane or a bridge or a building, some things about how it worked would change, as it got bigger or smaller.

Making things bigger or smaller changes the length of each part of those things, but also their area and volume. In this post, I want to get you thinking about these different ideas, because they help explain why ants aren't ten feet tall, why elephants and mice move so differently, and all kinds of other things.

The area of something tells you how much paint you would need to cover it. If you have a square piece of paper, you need a certain amount of paint to cover it. But if you get a square piece of paper that's twice as big on each side, you will need four times as much paint! You can see why it works this way in the picture below.

You can experiment with this. Cut out two squares, so that one is twice as big as the other on each side. Then, see how many pennies it takes to cover the smaller square, and how many to cover the larger square. Doubling the length of the square quadrupled the area.

Think about a very small house, maybe a doghouse. Maybe you only need one can of paint to paint this house. Imagine stretching that house, till it's twice as big on each side--you would need four cans of paint! If you stretched it until it was three times as big on each side, you'd need nine cans of paint. (Those numbers, 1,4,9,..., are called squares. Look at that picture again, and see if you can guess why.) That's a scaling effect. If you double the size of your house, you quadruple the number of cans of paint you need for it. If you triple the size of your house, you need nine times as many cans of paint!

The volume of something tells you how much paint it would take to completely fill it up. Since paint is kind of messy, you might prefer to think about it in terms of the number of marbles it would take to fill it up.

Let's think about that house, again. How much stuff could we put in the house? Let's suppose we have a house that's 10 feet on a side, and we fill the house up with marbles. Now, suppose we could somehow stretch the house to be 20 feet on a side. We'd need eight times as many marbles to fill it up! And if we stretched the house to be 30 feet on a side, we'd need twenty-seven times as many marbles! Look at the pictures to see why. (You can also probably guess that these numbers, 1,8,27,64,125..., are called cubes.)

You can do an experiment with this, too. You'll need to either find or make two boxes, so that the bigger box is twice as big as the smaller one on every side. Then, find something to fill the boxes with--packing peanuts will work, or ping pong balls. Count the number of items that fit in the smaller box. Then count the number in the larger box.

As anything gets bigger, its volume increases faster than its area, and its area increases faster than its height, width, or depth. Can you think of other ways things work differently, as they get bigger or smaller?

Ant Nebula
Photo credit: NASA/Space Telescope Science Institute
Note from Mama Joules to Itinerant Cryptopher: This was the biggest ant I could find! :)

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