At five o’clock, the water level in a pool reaches a height of 10 inches. At eight o’clock, the water level is at 50 inches. What is the rate of change of the water level per minute?
Well let’s try to visualize the problem. We’ve got this pool, and we’re topping it up with water. And at five o’clock the level is at 10 inches. But by eight o’clock, that level has reached the depth of 50 inches. So the level in that time is increasing from 10 inches to 50 inches; that is an increase of 40 inches. And the time that passed by, we went from five o’clock to eight o’clock, which is an increase of three hours.
But our question asked us for the rate of change of the water level per minute, so we need to convert those hours into minutes. Now each hour has got 60 minutes, so three hours has got three lots of 60 minutes, and three times 60 is 180, so that’s 180 minutes. So the water level is increasing at a rate of 40 inches in 180 minutes, but we want to know how much does the water increase every one minute.
So what do I have to do to 180 to turn it into one? Well I have to divide by 180. Now if I’ve only got 180th as much time, then the water level is only gonna increase 180th of the amount. So that’s 40 over 180 inches in one minute, so the water level is increasing at a rate of forty one hundred and eightieths of an inch per minute. But that’ll simplify; for start, if I divide the top by 10, I get four, and if I divide the bottom by 10, I get 18. And that again will cancel; if I divide four by two, I get two; if I divide 18 by two, I get nine. So it’s two-ninths of an inch per minute. So there’s our answer: the rate of change of the water level per minute is two-ninths of an inch per minute.