Question

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

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

Answer 1

Answer: (B)

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