The Leaking Tank Answers

Summary

How Long?

Suppose the tank is one foot tall. Then in 10 sec the filled tank's level goes down 3 in, at which point the pressure at the hole is less. Therefore it takes longer than 10 sec to fall another 3 in, making (b) the answer.

In general, let r (0 < r < 1) be the proportion of water remaining in the tank after some unit of time has elapsed, and call that time the tank's r-life. If the initial level A decreases to Ar between t = 0 and t = 1, then at any time t, any level α decreases to αr between t and t + 1. So between t = 1 and t = 2, the level drops from Ar to (Ar)r = Ar². From t = 0 to t = n, the level goes from A (= Ar⁰) to Arⁿ.

In this problem, the proportion remaining after 10 minutes is .75, so the amount left after another 10 minutes is :75² = .5625. Therefore after 20 minutes, the tank's level hasn't quite decreased to 50%, so again, the half-life is longer than 20 minutes.

Again: How Long?

The answer is (a). If after 30 sec the amount remaining is 50%, then after another 30 sec, the amount left is half of 50%, or 25%. That is, 75% of the water has been lost.