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Q.

Marbles of 1.4 cm diameter are dropped into a beaker containing some water and fully submerged. The diameter of the beaker is 7 cm. Find how many marbles have been dropped in it if water rises by 5.6 cm.


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a

250

b

450

c

150

d

350 

answer is C.

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Detailed Solution

We are given that the balls with a diameter of 1.4 cm are thrown into a beaker with a little water and completely submerged.
We know that balls are spherical in shape and the volume of a sphere is given by the relation
  V s = π d 3 6  .
IMG_256So we find the volume of one ball with a diameter of 1.4 cm.
⇒Vs= π (1.4) 3 6 c n 3  
Suppose that n balls are dropped into a beaker filled with water.
The cumulative volume of n balls will therefore be
V m =n× π 1.4 3 6 c m 3  
The diameter of the beaker is stated to be 7 cm. Let the height of the water level be h.
 The beaker is cylindrical in shape and we know that the volume of the beaker is given as
V b = π d 2 h 4  .
IMG_256
Volume of the beaker before the marbles were dropped are given as,
⇒ V b1 = π (7) 2 h 4 c m 3  
IMG_256After dropping n balls, the height increases by 5.6 cm. So the final height will be (h + 5.6) and the volume will be given as:
⇒ V b2 = π (7) 2 (h+5.6) 4 c m 3  
Therefore, the change in volume will be V b2 − V b1  .
V b2 − V b1 = π(7)(5.6) 4 c m 3  
Using Archimedes' principle, we can say that the change in volume of water will equal the total volume of the balls immersed in the water.
⇒ V b2 − V b1 = V m  
⇒ π (7) 2 (5.6) 4 c m 3 =n× π (1.4) 3 6 c m 3  
⇒ 7 × 7 ×(5.6) 4 =n× (1.4)× 1.4 × 1.4 6  
⇒2.8×3=n×(0.2)×(0.2)×(1.4) ⇒2×3=n×(0.04) ⇒ 6 0.04 =n ⇒n=150  
Therefore, the number of marbles is 150.
 
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