Question

A right circular conical vessel is being filled with green industrial waste at a rate of 60 cubic meters per second. How fast is the level rising after 150pi cubic meters have been poured in? the cone has a height of 60 m and a radius of 10 m at its brim.
The volume of a cone of height h and cross sectional radius r at its brim given by (1/3)(pi)r^2h

Answer 1

\[V = \frac{1}{3}\pi r^{2}h\]
\[r=h/6\]
\[V = \frac{1}{3*36}\pi h^{3}\]
\[h^3 = V\frac{108}{\pi}\]
\[h = [V\frac{108}{\pi}]^{\frac{1}{3}}\]
\[\frac{\partial{h}}{\partial{V}} = \frac{1}{3} [V\frac{108}{\pi}]^{\frac{-2}{3}}\]\[\frac{\partial{h}}{\partial{V}} = \frac{1}{3} [150*108]^{\frac{-2}{3}}\]

Answer 2

Actually forget the partials, and let's just look at the change w.r.t time
\[\frac{d{V}}{d{t}}=60\frac{m^3}{s}\] when
\[V = 150\pi m^3\]
\[\frac{d{h}}{d{t}} = \frac{1}{3}[V\frac{108}{\pi}]^{\frac{-2}{3}}\frac{108}{\pi}\frac{d{V}}{d{t}}\]
\[\frac{d{h}}{d{t}} = \frac{1}{3}[150\pi m^3\frac{108}{\pi}]^{\frac{-2}{3}}\frac{108}{\pi}60\frac{m^3}{s} = \frac{2160}{\pi}[150*108]^{\frac{-2}{3}}\frac{m^3}{m^2s}=1.074\frac{m}{s}\]

Answer 3

Thank you soooo much!!

Answer 4

You may need to check the calculation, but this is the idea.

Answer 5

Useful to let the units do the walking. (i.e. make sure each side of the equation has the same units)