Monday, February 14, 2011

Of Course You Can Do Caculus

You've just found this great condo high atop a twenty-story building overlooking the Charles River in Cambridge. The seller decides that she really likes you and she wants you to have the place, so much in fact, that she's willing to cut her price in half.

There's just one catch. The architect who designed the building was a bit eccentric and put a clause in her contract that required anyone purchasing the condo to install new carpeting in the living room and further, that the purchaser determine the amount of carpet required to cover the floor.

Normally, this wouldn't be too much of a challenge; you'd just measure one the length of the room (say twenty feet) and then measure the width of the room (say fifteen feet), multiply the two numbers together and get the area of the room in square feet (twenty times fifteen is three hundred).

Unfortunately, in this case, the computation is not so easy. The wall of the living room that faces the river is all glass (so far so good) and to optimize the view of the winding river, the facing wall winds with it as shown below.

You know how to calculate the area of squares and rectangles, but you've got no idea how to calculate the area of a shape that winds in all directions. As you contemplate the situation looking out over the river watching the crews rowing up and down and thinking about the great deal you've been offered you remember your first apartment in Cambridge (one with poor heating and cold floors) and how you carpeted it using discarded floor samples that a friend had found in a dumpster.

The multi-colored quilt you created from 2'x2' floor samples inspires you. If you were to lay out the floor samples in a way that covered as much as the floor as possible, then you could just count them up to determine approximately how many square feet you needed. Each 2'x2' floor sample covers 4 square feet.


In your mind's eye, you start laying out the floor samples counting as you go. Because the room is so oddly shaped, there's a lot of floor that you can't cover without bumping into walls, but that's OK; you're still getting a much better idea than you had before.

You complete your exercise counting 66 floor samples. So, you know that area of the room is about 66 * 4 square feet or 264 square feet.

The seller looks at you quite impressed, but says that unfortunately your number is not quite accurate enough.

Then you remember that in your first apartment, there were a couple of posts that passed from floor to ceiling. There was also an area that was raised to make room for some pipes. The place wasn't exactly rectangular and neither of the walls measured an even multiple of two. So, to fill the spaces, you had to cut the floor samples into smaller squares.



In your mind, with carpet squares that are now just 1'x1', you fill in the gaps adding 10 1'x1' squares. You add this to the first number and now have an even better approximation of 274 square feet.



Before the seller says anything, you think, "Wait, I can fill in even more space with 6"x6" squares (each representing 1/4 of a square foot). You fill in gaps counting 64 6"x6" carpet squares or an additional 16 square feet. You add this number to what you've counted so far and come up with 290 square feet.

The seller looks at you now very impressed saying, "The place is yours if you want it and by the way, you just invented Calculus."

You look at her a bit confused and she goes on to explain, "Actually, I designed this place and I never wanted it to belong to anyone who couldn't fully appreciate everything I did. So, I came up with my little 'test' to see how well potential buyers think. I can't tell you how many people have just walked out the door not knowing what to do when I ask them about the carpeting."



She continues saying, "Many people understand how to compute the areas of simple shapes like squares and rectangles, but they're completely baffled by more complex shapes. It seems never to occur to them that you can think about any complex shape as a collection of simple shapes. As you make the simple shapes smaller and smaller, the sum of their areas gets closer and closer to that of the complex shape. Calculus is just a set of formulas that let you calculate what you visualized using carpet squares; any complex shape is just a bunch of rectangles."


What have you invented not even knowing that you did it?

Happy Monday!
Teflon

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