Question

How fast is the water level dropping at the instant when the water is exactly 7.5 cm deep?

• A. $1/\pi$ cm/min
• B. $1/5\pi$ cm/min
• C. $1/2\pi$ cm/min
• D. $2/3\pi$ cm/min

Answer 1

The correct option is B $1/5\pi$ cm/min

If V is the volume of water and h, r are height and radius at time t

then $\frac{r}{h}=\frac{2}{3}$, $V=\frac{1}{3}\pi r^{2}h$

So $V=\frac{1}{3}\pi {\left(\frac{2h}{3}\right)}^2$ $h=\frac{4}{27}\pi h^3$

$\frac{dV}{dt}=\frac{4}{9}\pi h^2\frac{dh}{dt}$.

Putting $h=7.5$ and $\frac{dV}{dt}=-5$, we have $\frac{dh}{dt}=-\frac{1}{5\pi }$