A right circular conical vessel is being filled with green industrial waste at a rate of 75 cubic meters per second. How fast is the level rising after 250π cubic meters have been poured in? The cone has a height of 70 m and a radius 30 m at its brim. (The volume of a cone of height h and cross-sectional radius r at its brim is given by V = 1 3 πr2h. Round your answer to two decimal places.)

Question

A right circular conical vessel is being filled with green industrial waste at a rate of 75 cubic meters per second. How fast is the level rising after 250π cubic meters have been poured in? The cone has a height of 70 m and a radius 30 m at its brim. (The volume of a cone of height h and cross-sectional radius r at its brim is given by V =  1 3 πr2h. Round your answer to two decimal places.)

goformit100 · Answer

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dumbcow · Answer

the cone of water will be proportional to conical vessel 
$$\frac{r}{30} = \frac{h}{70}$$
$$r = \frac{3}{7} h$$
now define volume in terms of height(water level)
$$V = \frac{1}{3}\pi (\frac{3}{7}h)^{2}h = \frac{3}{49}\pi h^{3}$$
now using chain rule
$$\frac{dV}{dt} = \frac{dV}{dh}*\frac{dh}{dt}$$
given dV/dt = 75, solve for dh/dt
take derivative to find dV/dh
also find water level "h" when V = 250pi