Water Tank Problem

[puff2978]

A water tank has a circular base of
radius 4 meters and initally contains water to a depth
of 5 meters. Then water drains out at an increasing rate
so that after t hours the water is leaving at a rate of 6t m^3/hr
How many hours until the tank is empty ?

I know that Area*height is the volume of the cylinder...

 

[BigGlenntheHeavy]

$\displaystyle V_{tank} \ = \ \pi r^{2} h \ = \ \pi(16)(5) \ = \ 80\pi$

$\displaystyle \frac{dV}{dt} \ = \ -6t \ \implies \ \int dV \ =\int -6t dt \ \implies \ V(t) \ = \ -3t^{2}+C$

$\displaystyle V(0) \ = \ 80\pi \ = \ -3(0)+C, \ \implies \ C \ = \ 80\pi$

$\displaystyle Hence, \ V(t) \ = \ -3t^{2}+80\pi$

$\displaystyle Now \ when \ the \ tank \ is \ empty, \ V(t) \ = \ 0, \ ergo, \ 3t^{2} \ = \ 80\pi \ \implies \ t \ \dot= \ 9.1529 \ hrs.$

$\displaystyle Check: \ V(9.1529) \ = \ -3(9.1529)^{2}+80\pi \ \dot= \ .00068, \ good \ enough \ for \ government \ work.$