Calculating Drainage Time For A Cuboid Bath A Step-by-Step Guide

This article delves into a practical mathematical problem involving the calculation of drainage time for a cuboid-shaped bath. Understanding volume and flow rates are essential concepts in various real-world applications, and this example provides a clear illustration of how these principles can be applied. We will explore how to determine the volume of a cuboid, convert units appropriately, and ultimately calculate the time required to empty a bath given its dimensions and drainage rate. This comprehensive guide aims to provide a step-by-step solution, ensuring clarity and ease of understanding for readers of all mathematical backgrounds. We will cover the fundamental formulas, the conversion process from cubic meters to liters, and the final calculation of drainage time in minutes. By breaking down the problem into manageable steps, we aim to make the solution accessible and the underlying concepts readily graspable. Whether you are a student learning about volume and rates or simply someone interested in practical math applications, this article will offer valuable insights and a clear methodology for solving similar problems. This article is structured to provide a complete understanding of the problem-solving process. We will start by defining the given parameters, such as the dimensions of the bath and the drainage rate. Then, we will proceed to calculate the volume of the bath in cubic meters, which is a crucial step in determining the total amount of water it can hold. Next, we will convert the volume from cubic meters to liters, as the drainage rate is given in liters per minute. This conversion is necessary to ensure that we are working with consistent units. Once we have the volume in liters, we can calculate the time it will take to empty the bath by dividing the total volume by the drainage rate. This calculation will give us the drainage time in minutes, which is the final answer to the problem. We will also include explanations of the formulas and conversions used, making the article a useful resource for anyone looking to improve their understanding of volume, rates, and unit conversions. Throughout the article, we will use clear and concise language to ensure that the concepts are easily understood. We will also provide examples and illustrations to help visualize the problem and the solution. By the end of this article, you will have a solid understanding of how to calculate the drainage time for a cuboid bath, as well as the underlying mathematical principles involved.

Problem Statement

We are presented with a cuboid-shaped bath that has interior dimensions of 0.6 meters in width, 0.5 meters in depth, and 1.5 meters in length. The water drains from the bath at a rate of 5 liters per minute. The primary objective is to determine the time it will take for a completely full bath to empty, given these parameters. This problem requires a methodical approach, incorporating the calculation of the bath's volume and the conversion of cubic meters to liters. Understanding the problem statement is the first crucial step in solving any mathematical problem. In this case, we need to identify the key information provided and what the problem is asking us to find. The key information includes the dimensions of the cuboid bath (0.6 m, 0.5 m, and 1.5 m) and the drainage rate (5 liters per minute). The problem asks us to calculate the time it takes for the bath to empty completely. To solve this problem, we need to use the formula for the volume of a cuboid, convert the volume from cubic meters to liters, and then use the drainage rate to calculate the time. Breaking down the problem statement into these components helps us to develop a clear strategy for finding the solution. It is also important to understand the units of measurement used in the problem. The dimensions of the bath are given in meters, and the drainage rate is given in liters per minute. This means that we need to ensure that all our calculations are done using consistent units. If necessary, we may need to convert units to ensure consistency. For example, we may need to convert cubic meters to liters, as the drainage rate is given in liters per minute. This involves understanding the relationship between cubic meters and liters, which is a crucial part of solving the problem. By carefully analyzing the problem statement and identifying the key information and the units of measurement, we can develop a clear and effective strategy for finding the solution. This initial step is essential for ensuring that we approach the problem in a logical and organized manner. With a clear understanding of the problem statement, we can proceed to the next steps, which involve calculating the volume of the bath and converting units as necessary.

Step 1: Calculate the Volume of the Bath

To begin, we need to calculate the volume of the cuboid bath. The formula for the volume (V) of a cuboid is given by:

V = length × width × height

In this case, the length is 1.5 m, the width is 0.6 m, and the height (or depth) is 0.5 m. Substituting these values into the formula, we get:

V = 1.5 m × 0.6 m × 0.5 m

V = 0.45 m³

Therefore, the volume of the bath is 0.45 cubic meters. Calculating the volume is a fundamental step in determining how long it will take for the bath to empty. The volume represents the total amount of water the bath can hold, and it is essential for calculating the drainage time. The formula for the volume of a cuboid is straightforward and easy to apply, but it is crucial to ensure that the correct values are substituted into the formula. In this case, the length, width, and height of the bath are given in meters, so the resulting volume will be in cubic meters. It is also important to pay attention to the units of measurement and ensure that they are consistent throughout the calculation. If the dimensions were given in different units, such as centimeters, it would be necessary to convert them to meters before calculating the volume. The calculation of the volume is a crucial step in solving the problem, as it provides the basis for the subsequent calculations. Once we have the volume in cubic meters, we can proceed to convert it to liters, which is necessary because the drainage rate is given in liters per minute. This conversion will allow us to directly compare the volume of the bath with the drainage rate and calculate the time it will take to empty the bath. The accuracy of the volume calculation is critical for the overall accuracy of the solution. Any errors in the volume calculation will propagate through the subsequent steps and affect the final answer. Therefore, it is essential to double-check the calculations and ensure that the correct values are used. With the volume of the bath calculated, we can move on to the next step, which involves converting the volume from cubic meters to liters.

Step 2: Convert Cubic Meters to Liters

Since the drainage rate is given in liters per minute, we need to convert the volume of the bath from cubic meters to liters. The conversion factor is:

1 m³ = 1000 liters

To convert the volume of 0.45 m³ to liters, we multiply by 1000:

0. 45 m³ × 1000 liters/m³ = 450 liters

Thus, the bath has a volume of 450 liters. Converting cubic meters to liters is a crucial step in solving this problem because it allows us to work with consistent units. The drainage rate is given in liters per minute, so we need to express the volume of the bath in liters as well. The conversion factor between cubic meters and liters is a fundamental concept in unit conversions, and it is essential to understand this relationship to solve the problem accurately. The conversion factor of 1 m³ = 1000 liters is derived from the definition of a liter as the volume of a cube with sides of 10 centimeters. Since there are 100 centimeters in a meter, a cubic meter is equivalent to (100 cm)³ = 1,000,000 cubic centimeters. A liter is defined as 1000 cubic centimeters, so 1 cubic meter is equal to 1,000,000 cubic centimeters / 1000 cubic centimeters per liter = 1000 liters. Understanding this derivation can help to solidify the understanding of the conversion factor and its application. The conversion process involves multiplying the volume in cubic meters by the conversion factor of 1000 liters per cubic meter. This effectively changes the units from cubic meters to liters while maintaining the same volume. The resulting value of 450 liters represents the total amount of water the bath can hold, expressed in liters. This value is now directly comparable to the drainage rate, which is given in liters per minute. With the volume of the bath expressed in liters, we can proceed to the next step, which involves calculating the time it will take to empty the bath. This calculation will use the volume in liters and the drainage rate in liters per minute to determine the drainage time in minutes. The accuracy of the unit conversion is crucial for the overall accuracy of the solution. Any errors in the conversion process will propagate through the subsequent steps and affect the final answer. Therefore, it is essential to double-check the conversion calculation and ensure that the correct conversion factor is used. With the volume of the bath converted to liters, we can now calculate the drainage time.

Step 3: Calculate the Drainage Time

Now that we know the bath's volume is 450 liters and the water drains at a rate of 5 liters per minute, we can calculate the time it will take to empty the bath. The formula to calculate time is:

Time = Volume / Rate

In this case:

Time = 450 liters / 5 liters/minute

Time = 90 minutes

Therefore, it will take 90 minutes for the bath to empty completely. Calculating the drainage time is the final step in solving the problem. This step involves using the volume of the bath in liters and the drainage rate in liters per minute to determine the time it will take for the bath to empty completely. The formula for calculating time is a fundamental concept in rate problems, and it is essential to understand this relationship to solve the problem accurately. The formula Time = Volume / Rate is derived from the definition of rate as the amount of something that occurs per unit of time. In this case, the rate is the drainage rate, which is the amount of water that drains from the bath per minute. The volume is the total amount of water in the bath, and the time is the duration it takes for the water to drain completely. By dividing the volume by the rate, we can determine the time it takes for the event to occur. In this specific case, we are dividing the volume of the bath (450 liters) by the drainage rate (5 liters per minute) to find the time it takes for the bath to empty. The resulting value of 90 minutes represents the total time it will take for the bath to empty completely, given the drainage rate. This is the final answer to the problem, and it provides a practical solution to the question posed in the problem statement. The accuracy of the drainage time calculation depends on the accuracy of the volume calculation and the unit conversion. Any errors in these previous steps will propagate through to the final answer. Therefore, it is essential to ensure that all calculations are performed accurately and that the correct values are used. With the drainage time calculated, we have successfully solved the problem and can provide a clear and concise answer to the question posed in the problem statement. The solution demonstrates the application of fundamental mathematical concepts, such as volume calculation, unit conversion, and rate problems, in a practical real-world scenario.

Conclusion

In conclusion, by following a step-by-step approach, we have successfully determined that it will take 90 minutes for a completely full cuboid-shaped bath with dimensions 0.6 m by 0.5 m by 1.5 m to empty, given a drainage rate of 5 liters per minute. This problem highlights the practical application of mathematical concepts such as volume calculation, unit conversion, and rate problems. The importance of accurate calculations and unit conversions cannot be overstated, as any errors in these steps can significantly impact the final result. This exercise also demonstrates the importance of breaking down complex problems into smaller, more manageable steps. By first calculating the volume of the bath, then converting the volume to liters, and finally calculating the drainage time, we were able to solve the problem in a clear and logical manner. The ability to solve such problems is valuable in various real-world scenarios, from household tasks to engineering applications. Understanding the principles of volume, rate, and unit conversion is essential for making informed decisions and solving practical problems in everyday life. The solution to this problem also reinforces the understanding of the relationship between different units of measurement. The conversion between cubic meters and liters is a common conversion that is used in various fields, and this problem provides a practical example of how this conversion is applied. Furthermore, the problem illustrates the concept of rate and how it relates to time and volume. The drainage rate is a measure of how quickly the water is leaving the bath, and the drainage time is the total time it takes for the bath to empty. By understanding these concepts, we can solve various problems involving rates, such as calculating the time it takes to fill a tank or the speed of an object. In summary, this problem provides a valuable learning experience that reinforces fundamental mathematical concepts and their practical applications. By following a step-by-step approach and paying attention to accuracy and unit conversions, we can successfully solve complex problems and make informed decisions in various real-world scenarios. The ability to apply mathematical principles to practical problems is a valuable skill that can benefit individuals in various aspects of their lives.