A sphere of radius 14 cm is cut by a plane at a distance of x cm from

Question

A sphere of radius 14 cm is cut by a plane at a distance of x cm from its center, thereby breaking this sphere into two different pieces. If the cumulative surface area of these two pieces is 3/28 times more than that of the sphere, what is the value of x ? (The surface area of a sphere = 4πr^2)

A. 10.4
B. 11.4
C. 12.4
D. 12.5
E. 13

A. 10.4
B. 11.4
C. 12.4
D. 12.5
E. 13

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yashikaaggarwal · Answer

Closest answer in approximate value is C
Distance of X from centre is r= √(R^2-x^2)
The CSA of two pieces of square is equal to the 3/28 time more than the surface area of sphere
4πR^2+2πr^2=3/28 time more than the surface area of sphere.
4π14^2+2π√(R^2-x^2)^2=3/28
2464+2*22/7(14^2-x^2)=2728
X^2=154
X=12.4
OA is C

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GMATWhizTeam · Answer

Solutions 
 • radius of the sphere,  r = 14cms 
• So total surface area of the sphere  = 4*\pi*r^2 = 4*\pi*14^2 
• Now, the sphere is at cut by a plane at a distance x. Cutting of the sphere by a plane in two pieces will exposes two new circular surfaces, which will be identical. (shown in the below figure).
 https://www.dropbox.com/s/1204xwi59zsas0u/sphere_plus_pieces.png?dl=1 
• From the above figure, we can write, radius of each circular surface  = \sqrt{14^2 – x^2} 
• Now the increase the cumulative surface area of these two faces will only be contributed by these two identical circular faces.
 o Therefore,  2*\pi*(14^2 – x^2) = \frac{3}{28}*4*\pi*14^2 ⟹ 14^2 – x^2 = 42 ⟹x^2 =196 – 42 ⟹ x^2 = 154  
• Now,  12^2 = 144 < x^2 = 154 < 13^2 = 169 ⟹ 12 < x =\sqrt{154} < 13 
 o So, we can eliminate answer Options A, B and E. 
• Now, let’s check the remaining two answer options.
 o  Option D: 12.5 
   12.5 = \frac{25}{2 }⟹(12.5)^2 = \frac{625}{4} = 156.25   
o  Option C: 12.4 
   12.4 = \frac{124}{10} ⟹12.4^2 = (\frac{124}{10})^2 = \frac{15376}{100} = 153.76 
 This value is closer to 154 than the Option D (i.e. 156.25)   
Thus, the correct answer is  Option C.

poojakhanduja3017 · Answer

IMO C

The radius of the open surface left after dissecting it at x cm will be : \sqrt{14^2–x^2}
 We don't need to actually calculate the surface area of the two units, we can just play with the difference brought about by two new surfaces exposed . Remember that surface area of 2 hemispheres is 3*pi*r^2 multiplied by 2 ( and the increase is due to the open surface)
So, 
Step 1: 2*π*(14^2–x^2)=328∗4∗π∗14^2 
Step 2: 14^2–x^2=42
Step 3: x^2=196–42 = 154

This gives x = 12.4

sambitspm · Answer

The answer can be found out in the screenshot below.

bumpbot · Answer

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A sphere of radius 14 cm is cut by a plane at a distance of x cm from

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