2003 AIME II Problems/Problem 5

From AoPSWiki

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

2003 AIME II (Problems • Resources)
Preceded by Problem 4	Followed by Problem 6
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

Personal tools

Navigation

Search

Toolbox

2003 AIME II Problems/Problem 5

From AoPSWiki

Problem

Solution

See also