Thursday, February 25, 2010

Braintenance: Sand Sifting and Cypherin'

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Braintenance: Sand Sifting and Cypherin’

Here's a Braintenance question from over a week ago, with the answer posted right underneath it. I’ll bet you’d thought that I’d forgotten about it. Well…I never forget; I just get distracted.

Since we've become much more knowledgeable about cylinders and cones, this question 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?

As we recall:

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.

To get the answers, we just substitute the radius measurements and the amount of square footage of sand (both of which we’ve been given, thank goodness) into the formulas:

For the cylinder, it will be –
Pi x 3 feet^2 x height = 50 cubic feet, or
3.14 x 9 x height = 50, or
28.26 x height = 50, or
Height = 50/ 28.26 = 1.769 feet in height. (a short, squat cylinder…kind of stumpy, in fact)

For the cone, it will be –
1/3 x Pi x 9 feet^2 x height = 50 cubic feet, or
1/3 x 3.14 x 81 x height = 50, or
84.78 x height = 50, or
Height = 50/ 84.78 = 0.589 feet in height (a short, squat cone…even stumpier than the cylinder)

Observation = The height of the cone is one-third the height of the cylinder when the radius of the cone is three times that of the cylinder. Is this reciprocal relationship always true? Yes it is. In every case where the relationship between the cone's radius and the cylinder's radius is 1 to 3 (or 1:3), the ratio of the heights (assuming that they each contain the same volume of sand, or something more interesting) will be the inverse, or 3 to 1 (or 3:1).
My, but this gets me tired. I'm leaving. See you soon.

Douglas Castle
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