Tuesday, February 16, 2010

Braintenance: Sorting Sand Into Two Containers, Continued! (02.16.2010)

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Dear Friends:

Here is the question (if you have a negative outlook, you might call it a "problem" instead of a question) which I had posed to you last week: 

The mission is to find the heights of each two containers, one which is cylindrical and one of which is conical such that each one can hold the same exact amount (volume) of sand. Each of the containers must have a floor area (which is a circular base, naturally) of three square feet.

Here are the formulas required to either a) answer the question, or b) solve the problem. These formulas are part of the study of solid geometry:

Quick Note:  In the formulas that follow, instead of using superscript ("powers") to show exponents, I have used the ^ sign. For example, 5^2 is "five to the second power" or "five squared".

The Formula for the Area of a Circle = Pi x radius^2 [or Pi times the radius squared]

The Formula for the Volume of a Cylinder = Pi x radius^2 x height [or Pi, times the radius squared, times the height]

The Formula for the Volume of a Cone = 1/3 x Pi x radius^2 x height [or one-third of Pi, times the radius squared, times the height]

Pi is the Greek letter representing the constant ratio between the circumference of a circle and its diameter, which is often expressed as either 22/7, or as 3.14.


The first thing we know is that both containers have the same sized base (a circle, or more properly, a disc) and therefore have the same radius.

Just by observation, it is apparent that a cone having the same radius as a cylinder, would only hold (or contain) one third the amount of anything (such as sand) that it was filled with as would the cyclinder. Put another way, the cylinder can hold three times as much sand as the cone (assuming that they both have the same radius, which also means that they would have the same base size, or footprint). Seen from yet another perspective, the cone would have to be three times the height of the cylinder in order to hold (or contain) the same amount of a substance; in this case, to hold the same amount of sand.

The answer (or the solution) is that the height of the cone must be three times the height of the cylinder. For example, if the cylinder were three feet in height, the cone would have to be nine feet in height; if the cylinder were five feet in height, the cone would have to be fifteen feet in height.  There is an infinite number of possible correct numerical answers, providing that the cone is always three times as tall as the cylinder.

Here's a question for next time. Since we've become much more knowledgeable about cylinders and cones, this questions has two parts:

If we wanted to divide a truckload of 100 cubic feet of sand (note that we're working on sand, because it's easier to part the sand than to part the sea, generally speaking) equally into two containers, and one of them is a cylinder with a base (radius) of three feet, and the other is a cone with a radius of nine feet, how tall must the cone be? How tall must the cylinder be?

Take good care of your brain. Remember, your brain is like a muscle -- it grows stronger with exercise. Give it a workout. Take it out for a spin. Your mind can expand to meet most any challenge. Ironically, most of us are far more intelligent than we think we are.


Douglas Castle

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