Question

Water is poured into an inverted conical vessel of which the radius of the base is $2\, m$ and height $4 \,m$, at the rate of 77 litre/minute. The rate at which the water level is rising at the instant when the depth is $70\, cm$ is : (use $\pi=22 / 7$ )

• A. 10 cm/min
• B. 20 cm/min
• C. 40 cm/min
• D. none

Answer 1

Answer: (B)

$ \frac{ dV }{ dt }=77 \,lit / \min$
$\frac{ dh }{ dt }=\text { ? when } h =0.7 m $
$V =\frac{1}{3} \pi x ^2 h$
Also $\frac{ x }{ h }=\frac{2}{4} \Rightarrow h =2 x$
$V = \frac{1}{3} \pi \cdot \frac{ h ^2}{4} \cdot h =\frac{\pi h ^3}{12} \Rightarrow \frac{ dV }{ dt }=\frac{3 \pi h ^2}{12} \cdot \frac{ dh }{ dt } $
$77 \times 1000=\frac{\pi}{4} \times 70 \times 70 \cdot \frac{ dh }{ dt } $
$\Rightarrow \frac{ dh }{ dt }=20\, cm / \min$