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Intermediate

The marble box

Two marbles of sizes π and 4π are enclosed in a rectangular box. What is the total area of the box?

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Solution

The rectangle area is 24.

The first step is to extend the segment EC. The resulting right triangle AHC is similar to DCE. From the fact that the smaller circle has radius 1 and the bigger radius 2, it follows that their bases have proportion 2. Hence BE is ED and thus BHE and DCE are congruent.

Now it is a matter of filling in the lengths as shown and applying the Pythagorean theorem to arrive at x=4. Hence the original rectangle has size 4 x 6.

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