Question

General Application of Derivatives. Draw the necessary figure and indicate the dimension given.

1. A conical glass whose base radius is 5m and altitude 12m is being filled at the rate of. How fast is the surface rising when the liquid is 6m deep?

Answer 1

$r=5m,h=12m,\frac{dV}{dt}=3\\ \frac{r}{h}=\frac{5}{12}\\ r=\frac{5}{12}h\\ V=\frac{1}{3}\pi r^2h=\frac{25}{432}\pi h^3\\ \frac{dV}{dt}=\frac{25}{144}\pi h^2.\frac{dh}{dt}\\ 3=\frac{25}{144}\pi h^2.\frac{dh}{dt}\\ \frac{dh}{dt}=\frac{432}{25×\pi ×6^2}\\ \frac{dh}{dt}=\frac{12}{25\pi}$