Question

# 150 spherical marbles, each of diameter 1.4 cm are dropped in a cylindrical vessel of diameter 7cm containing some water, which are completely immersed in water. Find the rise in level of water in the vessel.

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Hint- Volume of increased level of water = total volume of 150 spherical marble.

Diameter of spherical marbles (d) =1.4cm
Radius of spherical marbles (r) =$\dfrac{d}{2} = \dfrac{{1.4}}{2} = 0.7cm$
Now the formula for volume of sphere ${v_1} = \dfrac{4}{3}\pi {r^3}$…………………. (1)
Putting the values in equation (1) we get
${v_1} = \dfrac{4}{3}\pi \times {(0.7)^3}$
Now the above is volume of 1 spherical marble but we have in total 150 marble
So volume of 150 marble
$\Rightarrow {v_1} = 150 \times \dfrac{4}{3}\pi \times {(0.7)^3}$
${v_1} = 216c{m^3}$
Now diameter of cylindrical vessel (D) =7cm
Radius of cylindrical vessel (R) =$\dfrac{D}{2} = \dfrac{7}{2} = 3.5cm$
Now Volume of cylinder ${v_2} = \pi {R^2}h$……………………….. (2)
Substituting the values in equation (2) we get
${v_2} = \dfrac{{22}}{7} \times {\left( {3.5} \right)^2} \times h = 38.5h{\text{ c}}{{\text{m}}^3}$
Now the volume of increased level of water = Total volume of 150 spherical marbles
$\Rightarrow {v_1} = {v_2} \\ \Rightarrow 38.5h = 216 \\ \Rightarrow h = 5.61{\text{ cm}} \\$
Rise of water level = 5.61 cm

Note- Whenever we face such a type of problem , the key concept that we need to keep in our mind is that the total volume of water that will eventually rise up in the cylindrical vessel will be due to the addition of the volume of entities that are immersed in the water.