Question

Water poured into an inverted conical vessel of which the radius of the base is 2 m and height 4 m, and the rate of 77 litres/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. $10cm/min$
• B. $20cm/min$
• C. $None$
• D. $40cm/min$

Answer 1

Answer: (B)

Given,

$\text{R}=2m=200cm$

$\text{H}=4m=400cm$

$\text{1}Litre=1000c{m}^3$

V=volume of Cone

$\frac{dV}{dt}=77\frac{litre}{minute}$

$\frac{dV}{dt}=77\times 1000\frac{c{m}^3}{minute}$

$\text{V}olumeofCone=\frac{1}{3}\times \frac{22}{7}\times r^{2}h$

Differentiating $V$ with respect to $t$ keeping $r$ as constant and $h$ as variable,

$\frac{dV}{dt}=\frac{1}{3}\times \frac{22}{7}\times r^2\times \frac{dh}{dt}$

$\frac{dh}{dt}=\frac{dV}{dt}\times 3\times \frac{7}{22}\times \frac{1}{r^2}$

$=77\times 1000\frac{c{m}^3}{minute}\times 3\times \frac{7}{22}\times \frac{1}{(200cm{)}^2}$

$=\frac{147}{80}\frac{cm}{min}$