Question

Coffee is draining from a conical filter with height and diameter both $15$ cm, into a cylindrical coffee pot with diameter $15$ cm. The rate at which coffee drains from the filter into the pot is $100$ cc/min. The rate in cm/min at which the level in the pot is rising at the instant when the coffee in the pot is $10$ cm, is

• A. $\frac{9}{16\pi }$
• B. $\frac{25}{9\pi }$
• C. $\frac{5}{3\pi }$
• D. $\frac{16}{9\pi }$

Answer 1

The correct option is D $\frac{16}{9\pi }$

For cylindrical pot,
Volume, $V=\pi r^{2}h$
$\frac{dV}{dt}=\pi (r^2\frac{dh}{dt}+h\cdot 2r\frac{dr}{dt})$
Since $r$ is constant,
$\therefore \frac{dr}{dt}=0$
$\frac{dV}{dt}=100=\pi r^2\frac{dh}{dt}$


$100=\pi \cdot \frac{225}{4}\cdot \frac{dh}{dt}(\because r=\frac{15}{2}\text{cm})$
$\Rightarrow \frac{dh}{dt}=\frac{400}{225\pi }=\frac{16}{9\pi }$ cm/min