Question

A closed conical vessel is filled with water fully and is placed with its vertex down. The water is let out at a constant speed. After $21 \,\min$, it was found that the height of the water column is half of the original height. How much more time in minutes does it require to empty the vessel?

• A. 21
• B. 14
• C. 7
• D. 3

Answer 1

Answer: (D)

Let $r$ and $h$ be radius and height of cone, respectively.

$\therefore$ Volume of cone $=\frac{1}{3} \pi r^{2} h$
Given, rate of outflow of water in $21 \,\min$, then $h$ changes to $\frac{h}{2}$
$\therefore$ Rate of outflow of water
$=\frac{1}{3}\left(\pi r^{2} h-\frac{\pi r^{2}}{4} \times \frac{h}{2}\right)=21 \,\min$
$\Rightarrow \frac{7 \pi r^{2} h}{8 \times 3}=21 \,\min$
$ \Rightarrow \frac{1}{3} \frac{\pi r^{2} h}{8}=3\,\min$
$\therefore$ Time required to empty vessels in $3 \,\min$.