Problem 4 (Draining a Tank)

Here is an interesting problem. Consider a tank with a square base and a height of 15 feet with a small hole at the bottom used for draining the tank. Suppose you know it takes hour to fill the tank when the hole at the bottom is plugged. Suppose you also know that when the plug is removed and water is poured into the tank (we always pour water into the tank at the same rate), it takes 5 minutes for the water to rise to 10 inches. How long will it take to empty a full tank by removing the plug ?

Solution

Since the tank has a square base and a height of 15 ft, we know the volume of the tank is 15 ft, where is the width of the base. And since the tank (the bottom plugged) can be filled in 1 hr (60 minutes), we have

b² cubic ft = (RATE IN cubic ft/min) (60 minute)

where RATE IN is the rate (cubic ft/min) at which we pour water into the tank. We also know that when the plug is removed and water is poured into the tank, the water level rises 10 inches (5/6 ft) in 5 minutes, and so we have

(5/6) b² cubic ft = 5 (RATE IN) - 5 (RATE OUT)

where RATE OUT is the rate at which water leaves the hole at the bottom of the tank. We can now eliminate b between the previous two equations, getting a relationship between the incoming water rate (RATE IN) and the outgoing rate (RATE OUT). We find

RATE IN = 3 (RATE OUT)

and since it takes 1 hr to fill the tank (with the hole plugged), it takes 3 hrs to empty the tank by pulling the plug.

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Last modified on Friday, February 12, 1999