2003 AIME II Problems/Problem 5

Problem

A cylindrical log has diameter $12$ inches. A wedge is cut from the log by making two planar cuts that go entirely through the log. The first is perpendicular to the axis of the cylinder, and the plane of the second cut forms a $45^\circ$ angle with the plane of the first cut. The intersection of these two planes has exactly one point in common with the log. The number of cubic inches in the wedge can be expressed as $n\pi$ , where n is a positive integer. Find $n$ .

Solution

The volume of the wedge is half the volume of a cylinder with height $12$ and radius $6$ . (Imagine taking another identical wedge and sticking it to the existing one). Thus, $V=\dfrac{6^2\cdot 12\pi}{2}=216\pi$ , so $n=\boxed{216}$ .

Video Solution by Sal Khan

Part 1: https://www.youtube.com/watch?v=EzE53aPGbrQ&list=PLSQl0a2vh4HCtW1EiNlfW_YoNAA38D0l4&index=5

Part 2: https://www.youtube.com/watch?v=yXg5CYuCcU4&list=PLSQl0a2vh4HCtW1EiNlfW_YoNAA38D0l4&index=25

- AMBRIGGS

2003 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 4	Followed by Problem 6
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2003 AIME II Problems/Problem 5

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Problem

Solution

Video Solution by Sal Khan

See also