This problem involves a constant rate of drainage from a tank. We can solve for both the initial volume of liquid and the time required for the tank to completely drain by assuming a linear relationship between time and volume.

Let’s define the variables:

• $V_0$ = initial volume of liquid in the tank (in liters).
• $V$ = volume of liquid in the tank at a given time (in liters).
• $t$ = time passed (in days).
• $r$ = constant rate of drainage (liters per day).

Step 1: Determine the rate of drainage

We are given:

• After 14 days, the volume is $2000$ liters.
• On a certain day (call it Day 0), the tank had $5000$ liters of liquid.

The rate of drainage $r$ is the change in volume divided by the change in time: $r = \frac{V_{\text{Day 0}} - V_{\text{Day 14}}}{\Delta t} = \frac{5000 - 2000}{14} = \frac{3000}{14} = 214.29 \text{ liters per day (approx)}.$ So, the tank drains approximately $214.29$ liters per day.

Step 2: Determine the initial volume of liquid

Now, to find the initial volume, we assume that the linear relationship holds from the initial day to Day 0. On Day 14, the volume is 2000 liters, so we can work backwards to find how much liquid was in the tank initially.

Using the formula: $V(t) = V_0 - r \cdot t$ At $t = 14$ days, $V(14) = 2000$, so: $2000 = V_0 - 214.29 \times 14$ $2000 = V_0 - 3000$ $V_0 = 5000 \text{ liters}.$ Thus, the initial volume of liquid in the tank was $5000$ liters.

Step 3: Determine how long it will take to completely drain the tank

To find when the tank will be completely empty, set $V(t) = 0$ in the linear equation: $0 = 5000 - 214.29 \cdot t$ Solving for $t$: $214.29 \cdot t = 5000$ $t = \frac{5000}{214.29} \approx 23.33 \text{ days}.$ Therefore, it will take approximately 23.33 days for the tank to completely drain.

Final Answers:

• The initial volume of liquid in the tank was 5000 liters.
• It will take approximately 23.33 days for the tank to completely drain.

Do you want more details or have any questions?

Here are 5 related questions you might explore:

1. What if the rate of drainage changed after a certain number of days?
2. How would the result change if the tank had been refilled partially after some days?
3. What would the time to drain be if the tank started with 6000 liters?
4. How would evaporation affect the rate of drainage?
5. What is the total volume drained after 10 days?

Tip: When solving linear word problems, always identify the relationship between variables early to set up the correct equation.