Filling a Tank with Variable Flow Rates: A Solution and Analysis

Have you ever encountered a situation where it took a different amount of time to fill a tank to different levels based on the remaining volume? If so, you’ve likely run into a scenario involving a variable flow rate. This article will explore a specific problem where half a tank takes 5 minutes to fill, but filling 2/3 of the remaining tank takes 30 minutes. We’ll break down the problem and provide a step-by-step solution, along with useful tips for dealing with similar scenarios.

The Problem Statement

Let’s start by examining the given data:

half of the tank takes 5 minutes to fill 2/3 of the remaining tank (which is 1/3 of the total tank) takes 30 minutes to fill

The key question is, how much time will it take to completely fill the tank?

Assumptions and Clarifications

Before diving into the solution, let’s clarify a few points to ensure we are solving the problem correctly:

Is the tank always empty when filling starts? If so, then the initial half-fill and subsequent fill are based on an empty starting point. Is the tank a regular shape with consistent flow properties (e.g., no obstructions or pressure changes)? If not, the problem might be more complex.

Analysis and Solution

Case 1: Tank is Regular Shape and Initially Empty

Assuming the tank is a regular shape and initially empty, we can use the given information to solve the problem step-by-step:

First, let’s determine the rate of flow per minute using the first piece of information:

Half of the tank takes 5 minutes to fill. Therefore, the flow rate (R) is:

R (1/2) tank / 5 minutes 1/10 of the tank per minute

Next, we use this flow rate to determine how long it takes to fill the remaining 1/3 of the tank (since 2/3 of the remaining 1/2 is 1/3):

The remaining 1/3 of the tank takes 30 minutes to fill at the same flow rate.

R (1/3) tank / 30 minutes 1/90 of the tank per minute, which is consistent with the previous calculation as 1/10 of the tank per minute is correct.

Now, since we know the flow rate is constant, we can calculate the total time to fill the tank:

The tank is divided into a 1/2 and a 1/3 section, with the remaining 1/6 being the last part to fill:

Total time to fill the tank 5 minutes (for 1/2) 30 minutes (for 1/3) (1/6) tank / (1/10) 5 30 1.67 approximately 41.67 minutes

Case 2: Tank is Initially Not Empty

Alternatively, if the tank is not initially empty, then the initial half-fill might not be based on an empty tank, complicating our calculations. In this case, we need to know the initial water level to proceed accurately. Without that information, the problem becomes indeterminate.

Conclusion

In conclusion, if the tank is a regular shape and initially empty, the total time to fill the tank is approximately 41.67 minutes. If the tank is not initially empty, the problem becomes more complex, and we need additional information to solve it accurately.

Additional Tips

Always ensure the tank shape is regular and the flow rate is constant for accurate calculations. If you suspect irregularities in flow or tank shape, use precise measurements and additional data to refine your calculations. For ongoing monitoring, use tools like flow meters or sensors to track the water level and flow rate accurately.