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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

A
r/√2
CORRECT ANSWER WRONG ANSWER
B
r/√3
CORRECT ANSWERWRONG ANSWER
C
r/√5
CORRECT ANSWERWRONG ANSWER
D
r/√6
CORRECT ANSWERWRONG ANSWER
Solution

Correct option is

(A)

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