Solving for the Capacity of a Tank in the Presence of Leaks and Inflows

The capacity of a tank can be determined when we have information about the leak rate and the inflow rate. Here we explore a scenario where a leak can empty a full tank in 8 hours, while a tap admits 6 liters per hour. When the tap is opened, the tank is emptied in 12 hours. We derive the capacity using systematic calculations.

Step 1: Determine the Leak Rate

Given that a leak can empty a full tank in 8 hours, the leak rate can be calculated as follows:

Leak (text{Rate} frac{C}{8}) liters per hour

Step 2: Determine the Net Effect when the Tap is Opened

When a tap is opened, it admits water at a rate of 6 liters per hour. The net rate at which the tank is being emptied when both the leak and the tap are considered is:

Net (text{Rate} text{Leak Rate} - text{Tap Rate} frac{C}{8} - 6) where (C) is the capacity in liters

Step 3: Calculate the Time to Empty the Tank with the Tap Open

When the tank is full and the tap is open, the tank is emptied in 12 hours. Therefore, the net rate can also be expressed as:

Net (text{Rate} frac{C}{12}) liters per hour

Step 4: Set Up the Equation

Setting the two expressions for the net rate equal to each other results in:

(frac{C}{8} - 6 frac{C}{12}) Solving for (C) gives:

First, eliminate the fractions by multiplying through by 24, the least common multiple of 8 and 12:

(24 left(frac{C}{8}right) - 24 times 6 24 left(frac{C}{12}right))

This simplifies to:

(3C - 144 2C)

Isolating (C) gives:

(3C - 2C 144)

(C 144) liters

Conclusion

The capacity of the tank is 144 liters.

Let's explore another scenario where the same principles are applied. A full tank is being emptied by a leak and at the same time being filled at a slower rate. It takes 15 hours to empty the tank in this way. In 10 hours one tank full is emptied by the emptying pipe. In the next 5 hours, half a tank more is emptied, which is supplied by the filling pipe in 15 hours.

Given that the filling rate of the first pipe is (frac{V}{9}) where (V) is the volume of the tank, the equation for the combined rate is:

(frac{V}{9} - frac{V}{X} frac{1}{12})

Solving for (X) results in:

(frac{V}{X} frac{1}{9} - frac{1}{12} frac{3}{108} frac{1}{36})

Therefore, the leak will empty the full tank in 36 hours.

Calculation with Given Flow Rates

At 8 liters per minute, which is 480 liters per hour, the quantity filled in 30 hours is 14,400 liters. This indicates that the tank's capacity is 14,400 liters.

The capacity of the tank is 14,400 liters.