Problema Solution
A plane P intersects a sphere with center O, forming a circle O'. If the plane is 8 inches from center O and the area of circle O' is 225pi square inches, find (in terms of pi) the number of cubic inches in the volume of the sphere.
Answer provided by our tutors
let
r = the radius of the circle O'
A = 225pi in^2 the area of circle O'
R = the radius of the sphere
V = (4/3)R^3 pi the volume of the sphere
first we will find r using the formula for the area or circle: A = r^2 pi
r^2 pi = 225 pi
r = 15 in
next using the Pythagorean Theorem we find the radius R of the sphere:
R^2 = 8^2 + r^2
R^2 = 64 + 225
R = 17 in
lets plug the value for R = 17 in in the formula for the volume of the sphere is V = (4/3)R^3 pi:
V = (4/3)17^3 pi
V = 2,929.33pi in^3
the volume of the sphere is 2,929.33 pi cubic inches.