Question

A cylinder contains water to the extent of ${\left(\frac{3}{4}\right)}^{th}$ of its height. The radius and height of the cylinder are $14$ cm and $28$ cm respectively. Four solid metal sphere balls of radii $2$ cm, $4$ cm, $6$ cm and $8$ cm are dropped into the cylinder. What is the maximum raise in the height of the water level after the balls are dropped in it?

• A. $15.55$
• B. $2.1$ cm
• C. $2.72$ cm
• D. None of these

Answer 1

Answer: (C)

$2.72$ cm

Volume of cylinder $=\pi r^{2}h=\frac{22}{7}\times (14{)}^2\times 28=17248c.c.$
Since water is filled up to $\frac{3}{4}$th of the height of the cylinder, the volume of water is cylinder $=\frac{3}{4}\times 17248=12936c.c.$
Volume of the four sphere are $\frac{32\pi }{3},\frac{256\pi }{3},\frac{864}{3}$ and $\frac{2048\pi }{3}$ respectively,
The total volume of the spheres $=\frac{32\pi +256\pi +864\pi +2048\pi }{3}=\frac{3200\pi }{3}c.c.$
Thus, on dropping the four balls into the cylinder, $\frac{3200\pi }{3}$ c.c. water would be displaced.
The height of the water level will rise by a volume of $\frac{3200\pi }{3}\therefore \pi r^{2}h=\frac{3200\pi }{3}$
$\Rightarrow 197h=\frac{3200}{6}$

Therefore, $h\approx 2.72$ cm