Question

A sphere has a radius 5.5cm. determine its volume and surface area. A frustum of a sphere is formed by two parallel planes, one through the diameter and the other at a distance h from the diameter. If the curved surface area of the frustum is to be 1/4of the surface area of the sphere, find the height h and the volume of the frustum.

Answer 1

for a sphere of radius R, just recall your basic formulas:
V = 4/3 πR^3
A = 4πR^2
for the frustrum, with small radius r and height h, it is easy to find the formulas for
curved area: 2πRh
volume: π/6 (3R^2 + 3r^2 + h^2)
so for this problem, with R=5.5, we have
2πRh = 1/4 * 4πR^2
h = R/2 = 2.75
Now we can find the smaller r, since a side view of the frustrum gives
r^2 = R^2 - h^2
so
V = π/6 (3R^2 + 3(R^2-h^2) + h^2)
= π/6 (6R^2 - 2h^2) = π/6 (6*5.5^2 - 2*2.75^2)
= 27.73π