Problem:
A cylindrical log has diameter 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 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 , where is a positive integer. Find .
Solution:
Let be a diameter of the circular face of the wedge formed by the first cut, and let be the longest chord across the elliptical face of the wedge formed by the second cut. Then is an isosceles right triangle and inches. If a third cut were made through the point on the log and perpendicular to the axis of the cylinder, then a second wedge, congruent to the original, would be formed, and the two wedges would fit together to form a right circular cylinder with radius inches and height . Thus, the volume of the wedge is , and .
The problems on this page are the property of the MAA's American Mathematics Competitions