Question

Water runs into an inverted conical tent at the rate of $20{ft}^3/min$ and leaks out at the rate of $5{ft}^3/min$. The height of the water is three times the radius of the water's surface. The radius of the water surface is increasing when the radius is $5ft$, is

• A. $\frac{1}{5\pi }ft/min$
• B. $\frac{1}{10\pi }ft/min$
• C. $\frac{1}{15\pi }ft/min$
• D. none of these

Answer 1

The correct option is A $\frac{1}{5\pi }ft/min$

Rate of change in volume of conical tent $=20-5=15{ft}^3/min$

Volume of cone $V=\frac{1}{3}\pi r^{2}h$

$V=\pi r^3$

$\Rightarrow \frac{dV}{dt}=3\pi r^2\frac{dr}{dt}$

$\Rightarrow \frac{dr}{dt}=\frac{1}{5\pi }ft/min$