The great applied problem introduction and brief history

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This problem came to me via a person living in the community in which I teach. A gentleman had a cylindrical tank that was lying horizontally on the ground. Its diameter was 14 feet and its length was 20 feet. The depth of the water in the tank was 4 feet. He wanted to know:

a) How many gallons of water were in the tank?

b) How many more gallons of water will it take to fill the tank?
At first I thought that this was a fairly trivial problem and that I would have his answers in a few minutes. However when I started to reason it out, it became apparent that the solution was much more involved. After I completed the solution, I realized that this problem had more mathematics interwoven in its solution than any other mathematics problem I have ever encountered. And the best part was that it was an actual, real-life problem! Hence The Great Applied Problem was born. A couple of days later I presented this problem to my Precalculus students and asked them to solve it. It was a wonderful journey through mathematics involving these varied mathematical concepts: the Pythagorean theorem, area of a triangle, area of a sector of a circle, area of a segment of a circle, right triangle trigonometry, area of a circle, volume of a cylinder, volume of a non-standard prism, and several instances of units conversion. The solution of the problem also requires the student to organize his/her work well and to be able to logically develop a plan of problem solving.

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