Question

A tank of capacity 25 litres has an inlet and an outlet tap. If both are opened simultaneously, the tank is filled in 5 minutes. But if the outlet flow rate is doubled and taps opened the tank never gets filled up. Which of the following can be outlet flow rate in litres/min?

• A. 2
• B. 6
• C. 4
• D. 3

Hint:

In pipes and cisterns problems, the phrase "never fills up" means the net filling rate is less than or equal to zero (\(\leq 0\)). This is a key translation from words to a mathematical inequality.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given two scenarios about filling a tank with an inlet and an outlet tap. We need to find a possible flow rate for the outlet tap based on these conditions.
Step 2: Key Formula or Approach:
Let the rate of the inlet tap be \(R_i\) litres/min and the rate of the outlet tap be \(R_o\) litres/min. The net filling rate when both are open is \(R_{net} = R_i - R_o\). The relationship between volume, rate, and time is: Volume = Rate \(\times\) Time. We will set up equations or inequalities based on the two given scenarios.
Step 3: Detailed Explanation:
Scenario 1: Both taps are opened. Tank capacity (Volume) = 25 litres.
Time to fill = 5 minutes.
Net rate = \(R_i - R_o\).
Using the formula: \[ 25 = (R_i - R_o) \times 5 \] Dividing by 5, we get: \[ R_i - R_o = 5 \quad \cdots(1) \] Scenario 2: The outlet flow rate is doubled. New outlet rate = \(2R_o\).
New net rate = \(R_i - 2R_o\).
In this case, the tank "never gets filled up". This implies that the net flow rate is either zero or negative (i.e., water is draining out or the inflow equals the outflow). So, we can write this as an inequality: \[ R_i - 2R_o \leq 0 \] \[ R_i \leq 2R_o \quad \cdots(2) \] Combining the results: From equation (1), we can express \(R_i\) in terms of \(R_o\): \[ R_i = 5 + R_o \] Now, substitute this expression for \(R_i\) into the inequality (2): \[ (5 + R_o) \leq 2R_o \] Subtract \(R_o\) from both sides: \[ 5 \leq 2R_o - R_o \] \[ 5 \leq R_o \] This inequality tells us that the outlet flow rate, \(R_o\), must be greater than or equal to 5 litres/min.
Checking the options: (A) 2: \(2<5\), so incorrect.
(B) 6: \(6 \geq 5\), so this is a possible value.
(C) 4: \(4<5\), so incorrect.
(D) 3: \(3<5\), so incorrect.
Step 4: Final Answer:
The only outlet flow rate from the options that satisfies the condition \(R_o \geq 5\) is 6 litres/min.