Question

A sphere of diameter 18 cm is dropped into a cylindrical vessel of diameter 36cm, partly filled with water. If the sphere is completely submerged, then the water level rises (in cm) by

• A. $3$
• B. $4$
• C. $5$
• D. $6$

Answer 1

The correct option is A $3$

Let $r$ & $R$ be the radius of the sphere and cylindrical vessel respectively.

Now, $2r=18cm\Rightarrow r=9cm$

$2R=36cm\Rightarrow R=18cm$

Let the rise in water level in the cylindrical vessel be ${}^{\prime }{h}^{\prime }$ cm.

Volume of sphere $=\frac{4}{3}\pi r^3$

Volume of liquid displaced in the cylindrical vessel $=\pi R^{2}h$

If the sphere is completely submerged in the vessel, then volume of liquid displace in the cylindrical vessel $=$ Volume of the sphere

$\therefore \pi R^{2}h=\frac{4}{3}\pi r^3\Rightarrow {\left(18\right)}^2\times h=\frac{4}{3}\times {\left(9\right)}^3$

$\Rightarrow h=\frac{4\times {\left(9\right)}^3}{3\times {\left(18\right)}^2}=3cm$

Thus, the rise in water level in the cylindrical vessel is $3cm.$