Question

A valve in a full 6000 gallon water tank is slowly opening. Water flows out of the tank through the valve. The flow rate in gallons per hour is given by the function f(t)=300t2 where t is in minutes. How much water flows out of the tank in the first 7 minutes? How many minutes does it take for the tank to empty completely?

Answer 1

In the first 7 minutes, you would expect 34300 gallons of water to flow out:

$\displaystyle\int_0^7f(t)\,\mathrm dt=300\int_0^7t^2\,\mathrm dt=100t^3\bigg|_{t=0}^{t=7}=100(7^3-0^3)=34300$

But the tank holds at most 6000 gallons, so the tank must get emptied in less than 7 minutes. So there is some times $t=c$ such that

$\displaystyle\int_0^cf(t)\,\mathrm dt=6000$

You have

$\displaystyle\int_0^cf(t)\,\mathrm dt=300\int_0^ct^2\,\mathrm dt=100c^3=6000$
$\implies c^3=60\implies c=\sqrt[3]{60}\approx3.915$