5

Part of 2003 AIME Problems

Problems(2)

Neighbourhood of a box

Source:

Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 3 by 4 by 5 units. Given that the volume of this set is

(m + n \pi)/p

m

n

p

are positive integers, and

n

p

are relatively prime, find

m + n + p

geometry3D geometryprismsphereAMCAIMErectangle

Cutting a Cylindrical Log

Source:

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,

n

is a positive integer. Find

n.

geometryrectangle