Question

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

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

Answer 1

Answer: (D)

For cylindrical pot
$V=\pi r^{2}h$
$\frac{dV}{dt}=\pi \left[r^2\frac{dh}{dt}+h\cdot 2r\frac{dr}{dt}\right]\left(r=\text{ constant, }\frac{dr}{dt}=0\right)$
hence, $100=\pi r^2\frac{dh}{dt}$
$100=\pi \cdot \frac{225}{4}\cdot \frac{dh}{dt}\left(r=\frac{15}{2}cm\right)$
$\frac{dh}{dt}=\frac{400}{225\pi }=\frac{16}{9\pi }cm/min$