Problem 18 - Elementary Algebra Within Reach

Question

Swimming Pool A swimming pool has a capacity of 10,800 gallons. A drain pipe empties the pool at a rate of 12 gallons per minute. How long, in hours, will it take for the pool to empty?

Step-by-Step Solution

Answer

It will take 15 hours for the pool to empty.

Step 1: Identify the given values

The total quantity, which is the capacity of the swimming pool, is 10,800 gallons. The rate, which is the rate at which the pool empties, is 12 gallons per minute.

Step 2: Calculate the time in minutes

Use the formula time = total quantity / rate to calculate the time in minutes. Substituting the given values into the formula: time = 10800 / 12 = 900 minutes.

Step 3: Convert the time from minutes to hours

Since there are 60 minutes in an hour, this can be done by dividing the total minutes by 60. So, time = 900 / 60 = 15 hours.

Key Concepts

Capacity and Rate Calculation

Understanding capacity and rate calculation is essential for solving a wide range of real-world problems, such as draining a swimming pool. In our example, the capacity of the pool is the total volume of water it holds, which is 10,800 gallons. The rate refers to the speed at which the pool is being emptied, in this case, 12 gallons per minute.

To find out how long it will take to empty the pool, we need to divide the capacity by the rate. The formula looks like this: \[ \text{{Time}} = \frac{{\text{{Total Capacity}}}}{{\text{{Rate}}}} \.\] By plugging in the numbers, we obtain: \[ \text{{Time}} = \frac{{10800}}{{12}} = 900 \text{{ minutes}}.\] This type of calculation is critical in various settings, not just for swimming pools but for any scenario where you need to consider how much of something can be processed in a given time frame.

Unit Conversion

Once we have calculated time in a specific unit, we may need to convert it to another unit for practical reasons. In our swimming pool problem, we initially calculate the time in minutes. However, expressing time in minutes may not always be convenient, especially for longer durations. Therefore, converting minutes to hours simplifies our understanding and communication of the time required for the task.

To convert from minutes to hours, we need to remember that there are 60 minutes in an hour. The conversion formula is: \[ \text{{Time in hours}} = \frac{{\text{{Time in minutes}}}}{{60}} \.\] Hence, for our problem: \[ \text{{Time in hours}} = \frac{{900}}{{60}} = 15 \text{{ hours}}.\] This essential skill of unit conversion appears frequently not only in academic problems but also in everyday life such as cooking, travel, and science.

Algebraic Problem-Solving

Effective algebraic problem-solving requires an understanding of how to manipulate equations and formulas to find an unknown variable, often represented by letters such as \( x \), \( y \), or \( t \) for time. In the context of our swimming pool scenario, our unknown is the time taken to empty the pool.

Following a series of algebraic steps, we express the time as a function of the pool's capacity and the rate of drainage. Algebra not only helps us to solve equations but also teaches us to approach problems in a structured and logical manner. By isolating the variable, we simplify the problem into a manageable calculation, as demonstrated in our earlier steps where we divide the total capacity by the rate and then convert minutes to hours. Algebraic thinking is a fundamental tool that powers many fields, including engineering, economics, and informatics.