How would you solve a general problem of a steady stream of leakage:
Water leaks at a rate
$$r(t)= 20 \sqrt{3} \sec^2 (2t) \, \frac{\text{gallons}}{\text{hour}}.$$
At time 0, there are 50 gallons of water.
So how should I find a function that represents the amount remaining at a certain time, say $\pi/6$ hours?
I understand to take the integral of $r(t)$, but how would you proceed then?
As a follow up question, how long will it take the sludge to leak completely? I got an answer of 2x = infinity…
Best Answer
Hint: You are given the if $W(t)$ is the amount of water in the tank, your function $r(t)$ is $W'(t)$. So if you integrate $$ \int W'(t) dt=\int r(t) dt $$ that will give you $W(t)$, the amount of water at time $t$. Use the fact that $W(0)=50$. Then the question is just asking what is $W(\frac{\pi}{6})$.