Question

A cylindrical vessel is filled with water up to some height. If a sphere of diameter 8 cm is dropped into the cylinder the water level rises by half of the initial level. Instead, if a second sphere of diameter 16 cm is dropped the water level rises to a height $h_2$. What percentage of this new height $h_2$ to the initial level of water?

• A. 40 %
• B. 20%
• C. 15%
• D. 16%

Answer 1

The correct option is B

20%

Let h be the initial height of the cylinder
when we immerese first sphere in cylinder,
volume of the water displaced =volume of the sphere immersed
we get $\pi r^2\left(0.5h\right)=\left(\frac{4}{3\pi }\right)(4{)}^3....(1)$
when second sphere of radius 8 cm is immersed in cylinder, change in volume
volume of the water displaced =volume of the sphere immersed
$\pi r^2(h_2-h)=\left(\frac{4}{3\pi }\right)(8{)}^3...(2)$
Dividing equation (1) & (2) we get
$\frac{\pi r^2(h_2-h)}{\pi r^2\left(0.5h\right)}=\frac{\frac{4}{3}\pi 8^3}{\frac{4}{3}\pi 4^3}$
$h_2-h=4h{h}_2=5h$
% of $h_2$ its initial height =$\left(\frac{1}{5}\right)\times 100\%=20\%$