Question

Water is leaking from a conical funnel at the rate of $5c{m}^3/\sec$. If the radius of the base of the funnel is $5cm$ and the altitude is $10cm$, find the rate at which the water level is dropping when it is $2.5$ cm from the top.

Answer 1

$\begin{array}{c}\frac{dv}{dt}=5c{m}^3/\sec \\ \tan \theta =\frac{r}{h}=\frac{5}{10}\Rightarrow h=2r\\ v=\frac{1}{3}\pi \times r^{2}h=\frac{1}{3}\pi \times {\left(\frac{h}{2}\right)}^{2}h=\frac{\pi h^3}{12}\\ \frac{dv}{dt}=\frac{\pi }{12}\times 3\times h^2\frac{dh}{dt}=5\\ h=10-2.5=7.5cm\\ 5=\frac{1}{3}\pi \times (7.5{)}^2\frac{dh}{dt}\\ \frac{dh}{dt}=\frac{20}{{\left(7.5\right)}^2\times \pi }\end{array}$