Question

A conical vessel of radius $6cm$ and height $8cm$ is completely filled with water. A metal sphere is now lowered into water. The size of the sphere is such that when it touches the inner surface, it just gets immersed. The fraction of water that overflows from the conical vessel is

• A. $\frac{3}{8}$
• B. $\frac{5}{8}$
• C. $\frac{7}{8}$
• D. $\frac{5}{16}$

Answer 1

Answer: (A)

$\frac{3}{8}$

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.
So, Volume of water in the conical vessel $=\frac{1}{3}\times \frac{22}{7}\times 6^2\times 8{cm}^3$
Since the sphere touches the inner surface when it gets immerse, its diameter $=6cm$
=> its radius $=3cm$
Volume of water that overflows $=$ Volume of the sphere $=\frac{4}{3}\pi r^3=\frac{4}{3}\times \frac{22}{7}\times 3^3{cm}^3$
Fraction of water that overflows $=\frac{\frac{4}{3}\times \frac{22}{7}\times 3^3}{\frac{1}{3}\times \frac{22}{7}\times 6^2\times 8}=\frac{3}{8}$