Questions & Answers

Question

Answers

Answer

Verified

47.5k+ views

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.

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.

Students Also Read