An Easy AIME Problem!

Geometry Level 4

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 have exactly one common point in common with the log. The number of cubic inches in the wedge can be expressed as $n\pi$ . Find $n$ .

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This is in the set Contest Problems
This is an AIME Problem.

The answer is 216.

2 solutions

Daniel Liu
Aug 6, 2014

Note that two of these wedges put together forms a cylinder with diameter $12$ and height $12$ . The volume of this cylinder is $12\cdot 6\cdot 6\cdot \pi=432\pi$ so the area of one of the wedges is $432\pi\div 2 = \boxed{216}\pi$ .

Very short yet understandable. You never fail to amaze me Mr. Daniel Liu!

Sean Ty - 6 years, 10 months ago

Same solution.

Niranjan Khanderia - 6 years ago

How can you get height 12 if nothing about height was mentioned in the givens..? Was I missing a word that should've given the piece of info? I would also like to know how I'm just supposed to guess the exact orientation of the planes cutting into the cylinder....It just says perpendicular to the axis (unless this means a central axis)...So I'm confused on drawing this problem because of the wording

chris recupero - 4 years, 1 month ago

45-45-90 triangles.

alex wang - 1 year, 4 months ago

Munem Shahriar
Oct 11, 2017

The volume of the wedge is half the volume of a cylinder with height $12$ and radius $6$ .

So, $V=\dfrac{6^2\cdot 12\pi}{2}=216\pi$ , hence $n=\boxed{216}$ .

An Easy AIME Problem!

The answer is 216.

2 solutions

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