Pizza and Problems/Pizza and Problems 1 (9/12/08)
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Use this page to post solutions to our seventh Pizza and Problems session, held 9/12/08. Examine the original problem set.
A great time was had by all at our meeting of Pizza and Problems. We had some very cool problems, good pizza, and enthusiam and chuckles were high. Finally, we had some very good student presentations of solutions. Some students worked together on problems, arguing their case and developing ideas! A very cool strategy.
Contents |
Student Solutions to the Problems in Pizza and Problems Set #1
Problem 1: Solution by:
Problem 2: Solution by:
Problem 3: Solution by:
Problem 4: Solution by:
Problem 5: Solution by: Bruce Wagner
The following picture shows an upside-down, perspective view of the situation. Lengths are measured in thousands of feet. Point E is the underground point where the oil well struck pay sand. Point F corresponds to the point on the ground directly above point E.
By the Pythagorean Theorem, m2 + w2 = 212, m2 + x2 = 62, m2 + y2 = u2, and m2 + z2 = 182. Subtracting the first two equations yields w2 − x2 = 212 − 62, and subtracting the last two equations yields y2 − z2 = u2 − 182.
Now consider the diagram below showing the situation on the ground. Draw perpendiculars from point F to the top and bottom of the rectangle as shown. Again, by the Pythagorean Theorem, 
and 
. It follows that 
.
Therefore, 212 − 62 = w2 − x2 = y2 − z2 = u2 − 182, so 212 − 62 = u2 − 182. Solving this last equation for u yields u = 27. i.e., 27000 feet.
The useful formula referred to in the problem can be seen by rewriting the final equation as a sum on both sides: 212 + 182 = u2 + 62. In other words, for a pyramid with a rectangular base, the sums of the squares of opposite lateral edges are equal.
