Understanding the Cistern Pipe Problem

This problem involves calculating the total capacity of a cistern (a tank) based on how long different pipes take to fill or drain it, and how much a waste pipe drains per minute.

We have two pipes that fill the cistern and one waste pipe that drains it. When all three are working together, the cistern fills up in a specific amount of time. We are given the rates for the filling pipes and the draining pipe.

Setting Up Rates for Pipes

Let's denote the total capacity of the cistern as $C$ litres.

Pipe 1 fills the cistern in 36 minutes. This means in 1 minute, Pipe 1 fills $\frac{1}{36}$ of the cistern. So, its filling rate is $\frac{C}{36}$ litres per minute.

Pipe 2 fills the cistern in 45 minutes. In 1 minute, Pipe 2 fills $\frac{1}{45}$ of the cistern. So, its filling rate is $\frac{C}{45}$ litres per minute.

The waste pipe drains 40 litres per minute. Its draining rate is 40 litres per minute.

Calculating Combined Filling Rate

When all three pipes are opened simultaneously, the net filling rate is the sum of the filling rates minus the draining rate.

Net filling rate per minute = (Rate of Pipe 1) + (Rate of Pipe 2) - (Rate of waste pipe)

Net filling rate per minute = $\left(\frac{C}{36} + \frac{C}{45} - 40\right)$ litres per minute.

Using the Total Time to Fill

We are told that when all three pipes are opened, the cistern fills in one hour. One hour is equal to 60 minutes.

The total capacity of the cistern ($C$) is equal to the net filling rate multiplied by the time taken to fill it.

Total Capacity = (Net filling rate per minute) $\times$ (Time in minutes)

$C = \left(\frac{C}{36} + \frac{C}{45} - 40\right) \times 60$

Solving for the Cistern Capacity

Now, we need to solve this equation for $C$ to find the capacity of the cistern.

First, distribute the 60 on the right side of the equation:

$C = \frac{C}{36} \times 60 + \frac{C}{45} \times 60 - 40 \times 60$

$C = \frac{60C}{36} + \frac{60C}{45} - 2400$

Simplify the fractions $\frac{60}{36}$ and $\frac{60}{45}$:

• $\frac{60}{36}$ can be simplified by dividing both by 12: $\frac{60 \div 12}{36 \div 12} = \frac{5}{3}$
• $\frac{60}{45}$ can be simplified by dividing both by 15: $\frac{60 \div 15}{45 \div 15} = \frac{4}{3}$

Substitute these simplified fractions back into the equation:

$C = \frac{5C}{3} + \frac{4C}{3} - 2400$

Combine the terms with $C$ on the right side:

$C = \left(\frac{5}{3} + \frac{4}{3}\right) C - 2400$

$C = \left(\frac{5 + 4}{3}\right) C - 2400$

$C = \left(\frac{9}{3}\right) C - 2400$

$C = 3C - 2400$

Now, rearrange the equation to solve for $C$. Subtract $C$ from both sides:

$0 = 3C - C - 2400$

$0 = 2C - 2400$

Add 2400 to both sides:

$2400 = 2C$

Divide by 2:

$C = \frac{2400}{2}$

$C = 1200$

The capacity of the cistern is 1200 litres.

Final Answer

The capacity of the cistern is 1200 litres.

Revision Table: Cistern Pipe Calculation

Additional Information: Work and Time Concepts

Problems involving pipes filling or emptying tanks are similar to work and time problems. The amount of work done is analogous to the volume of the tank, and the time taken corresponds to the time period. The rate of work is the amount of tank filled or emptied per unit of time.

• If a pipe fills a tank in $t$ hours, its filling rate is $\frac{1}{t}$ of the tank per hour.
• If a pipe empties a tank in $t$ hours, its emptying rate is $\frac{1}{t}$ of the tank per hour.
• When multiple pipes work together, their rates are added (for filling) or subtracted (for emptying) to find the combined rate.
• If the combined rate is positive, the tank fills. If it's negative, the tank empties.
• Total work = Combined Rate $\times$ Time. In this case, Total Capacity = Net Rate $\times$ Time.