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

Mensuration

ICSE

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Answer

Given,

Diameter of spherical marbles = 1.4 cm

Radius (r) = 1.42\dfrac{1.4}{2} = 0.7 cm.

Diameter of beaker = 7 cm

Radius of beaker (R) = 72\dfrac{7}{2} = 3.5 cm

Increase in water level (h) = 5.6 cm.

Let n marbles are dropped.

⇒ Volume of water increased in beaker = n × Volume of one marble (sphere)

⇒ πR2h = n×43πr3n \times \dfrac{4}{3}πr^3

⇒R2h=n×43r3⇒n=3R2h4r3⇒n=3×(3.5)2×5.64×(0.7)3⇒n=3×12.25×5.64×0.343⇒n=150.\Rightarrow R^2h = n \times \dfrac{4}{3}r^3\\[1em] \Rightarrow n = \dfrac{3R^2h}{4r^3} \\[1em] \Rightarrow n = \dfrac{3 \times (3.5)^2 \times 5.6}{4 \times (0.7)^3} \\[1em] \Rightarrow n = \dfrac{3 \times 12.25 \times 5.6}{4 \times 0.343} \\[1em] \Rightarrow n = 150.

Hence, 150 marbles are dropped in the beaker.

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