Question

A conical vessel of radius $12cm$ and depth $16cm$ is completely filled with water. A sphere is lowered into the water and its size is such that when it touches the inner curved surface of the vessel, it is just immersed up to the topmost point of the sphere. How much water over flows out of the vessel out of the total volume $V$ cubic units?

• A. $\frac{1}{4}V$
• B. $\frac{3}{4}V$
• C. $\frac{3}{8}V$
• D. $\frac{8}{9}V$

Answer 1

The correct option is D $\frac{8}{9}V$

Diameter of the sphere is equal to the height of the cone $=16cm$
=> Radius of the sphere $=8cm$
Volume of a cone $=\frac{1}{3}\pi r^{2}h$

where, $r$ is the radius of the base of the cone and $h$ is the height.
Hence, volume of the given cone,

$V=\frac{1}{3}\pi \times {12}^2\times 16=768\pi$
Volume of sphere $=\frac{4}{3}\pi r^3$
Hence, Volume of water displaced $=$ Volume of the sphere $=\frac{4}{3}\times \pi \times 8^3=\frac{2048}{3}\times \pi$
Now, $\frac{\frac{2048}{3}\times pi}{768\times \pi }=\frac{8}{9}$
Hence, $\frac{8}{9}$ of the total volume is spilled out.