Exercise 10

Suppose that a cylindrical water tank with a hole in the bottom is filled with water. The water, of course, will leak out and the height of the water will decrease. Let $h(t)$ denote the height of the water. A physical principle called Torricelli's Law implies that the height decreases at a rate proportional to the square root of the height.

1. Express this fact using $k$ as the constant of proportionality.

2. Suppose you have two tanks, one with $k=-1$ and another with $k=-10$. What physical differences would you expect to find?

3. Suppose you have a tank for which the height decreases at $20$ inches per minute when the water is filled to a depth of $100$ inches. Find the value of $k$.

4. Solve the initial value problem for the tank in part (c), and graph the solution you determine.

5. How long does it take for the water to run out of the tank?

6. Is the solution that you found valid for all time $t$? If so, explain how you know this. If not, explain why not.