Understanding the Impact of Pipes and Holes on Tank Filling Time

Introduction

When dealing with tanks that are being filled through multiple pipes, the efficiency of the process can be significantly affected by either additional holes or valves that can drain the liquid. This article explores a problem where two pipes fill a tank and how the presence of a small hole increases the filling time. We will derive the time taken for the hole to empty the tank entirely.

Problem Statement

Two pipes can independently fill a tank in 20 hours and 30 hours, respectively. The presence of a small hole makes it take 3 more hours to fill the tank. The goal is to determine how long it would take for the hole to empty the tank completely.

Calculating the Filling Rates of the Pipes

The filling rate of the pipes is determined as follows: Let the filling rate of Pipe A be RA 1/20 tanks per hour. Let the filling rate of Pipe B be RB 1/30 tanks per hour.

Combined Filling Rate

When both pipes work together, their combined rate (Rcombined) is calculated as:

Rcombined RA RB

Substituting the values:

Rcombined 1/20 1/30

Both fractions must be added to a common denominator (60):

1/20 3/60

1/30 2/60

Rcombined 3/60 2/60 5/60 1/12 tanks per hour

Effect of the Hole on Filling Time

With the hole present, it takes 3 more hours to fill the tank. Therefore, the effective filling time with the hole is 15 hours (12 hours without the hole plus 3 additional hours).

Determining the Hole's Emptying Rate

Let r be the emptying rate of the hole in tanks per hour. The effective rate of filling when the hole is present is given as 1/15 tanks per hour. Using this, we can set up an equation to find r:

Effective Rate Combined Rate - Hole's Rate

1/15 1/12 - r

Solving for r:

r 1/12 - 1/15 5/60 - 4/60 1/60 tanks per hour

The rate of the hole is 1/60 tanks per hour, meaning it takes 60 hours to empty the tank completely.

Conclusion

The presence of a small hole can severely impact the efficiency of filling a tank. In this case, it takes 60 hours for the hole to empty the tank. Understanding rates and their combinations can help in managing such scenarios. Knowing the individual filling rates and the effect of the hole allows for better planning and resource allocation in various industries, including water management and industrial processes.

Keywords

Tank filling, pipe rate, hole effect, emptying time.