# CAT Practice : Mensuration

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Cutting a sphere into two equal parts increases its surface area by 50%, what will be the increase if we cut it in some other fashion?

## Spheres

Q.2: 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.
1. ${r \over \sqrt{2}}$
2. ${r \over \sqrt{3}}$
3. ${r \over \sqrt{5}}$
4. ${r \over \sqrt{6}}$

Choice A. ${r \over \sqrt{2}}$

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## Detailed Solution

Area = 4πr2

Cumulative area of the two pieces = 25% more than the by square.
Area of 2 pieces = 1.25 × 4π2 = 5πr2
Extra area = πr2
Extra area = area of two new circles that are now created circles.
Area each new circle = ${{\pi}r^2 \over 2}$

Let radius of new circle be r1.
Now, πr12 = ${{\pi}r^2 \over 2}$
r1 = ${r \over \sqrt{2}}$

Now, r1, h and r form a right angled triangle.
h2 + r12 = r2
h2 + ${\left({r \over \sqrt{2}}\right)^2}$ = r2
h = ${r \over \sqrt{2}}$

Correct Answer: ${r \over \sqrt{2}}$

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## More questions from Mensuration

Mensuration is the science of measurement. Cylinders, cones, cuboids, rectangular parallelopipeds etc.