Question
A sphere of radius r is cut by a plane at a distance of h from its center, thereby breaking this sphere into two different pieces. The cumulative surface area of these two pieces is 25% more than that of the sphere. Find h
CAT 2021
Correct option is
Explanatory Answer :
When a sphere of radius r is cut by a plane at a distance h from its centre; total curved surface area of the two pieces remains same as the curved surface area of the mother sphere {4 * π * (r^2)}.
As a consequence of this cut, two plane surface areas are generated each being circular in shape with radius √{(r^2) - (h^2)}.
Thus, additional plane surface area generated due to this cut = 2 * π * {(r^2) - (h^2)}, which must be 25% of the total curved surface area of the mother sphere.
As per given condition,
[2 * π * {(r^2) - (h^2)}] / {4 * π * (r^2)} = 25 / 100
{(r^2) - (h^2)} / (r^2) = 1 / 2
2 * (r^2) - 2 * (h^2) = (r^2)
2 * (h^2) = (r^2)
(h^2) = (r^2) / 2
