Intr-un Vas Cubic Cu Latura De 0,75 M, Se Toarnă 225 Litri De Apă. Care Este Înălțimea La Care Se Ridică Apa? Opțiunile Sunt A) 0,2 M; B) 0,8 M; C) 0,4 M; D) 0,75 M.

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Introduction

This article addresses a common problem in geometry: determining the height of water in a cubic vessel after a certain volume of water is poured in. The problem presented involves a cubic vessel with a side length of 0.75 meters, into which 225 liters of water are poured. The goal is to calculate the height to which the water rises inside the cube. This problem combines concepts of volume, unit conversion, and basic geometry, making it an excellent exercise for students and anyone interested in practical applications of mathematics. We will walk through the necessary steps, including converting units, calculating the volume of the cube, and finally, determining the water level. Understanding these steps can help in various real-world scenarios, such as calculating liquid levels in tanks or containers.

Understanding the Problem

Before diving into the calculations, it’s crucial to clearly understand the problem statement. We have a cubic vessel, which means all its sides are equal in length. The side length is given as 0.75 meters. The vessel is filled with 225 liters of water, and our task is to find the height to which the water rises. This involves converting liters to cubic meters, calculating the volume of the cube, and using this information to find the height of the water level. The key concepts here are volume, unit conversion, and the formula for the volume of a cube. To solve this problem efficiently, we need to ensure that all units are consistent. We'll convert liters to cubic meters to match the meter unit of the cube's side length. This conversion is essential for accurate calculations and a correct final answer. Remember, a cubic vessel's volume calculation relies on the formula side * side * side, and understanding this is crucial for determining the water height.

Step 1: Convert Liters to Cubic Meters

The first step in solving this problem is to convert the volume of water from liters to cubic meters. This conversion is necessary because the dimensions of the cube are given in meters, and we need to work with consistent units. The conversion factor between liters and cubic meters is: 1 cubic meter (m³) = 1000 liters. Therefore, to convert 225 liters to cubic meters, we divide 225 by 1000:

225 liters / 1000 = 0.225 cubic meters (m³)

This conversion is crucial because it allows us to work with the same units throughout the problem. By expressing the volume of water in cubic meters, we can directly compare it to the volume of the cubic vessel, which will also be calculated in cubic meters. This step ensures that our calculations are accurate and that we arrive at the correct answer. Using the correct conversion factor is a fundamental skill in many scientific and engineering calculations, making this step particularly important. Remember, accurate unit conversion is the cornerstone of accurate problem-solving in math and physics.

Step 2: Calculate the Volume of the Cubic Vessel

Next, we need to calculate the total volume of the cubic vessel. The formula for the volume of a cube is:

Volume = side × side × side

Given that the side length of the cube is 0.75 meters, we can plug this value into the formula:

Volume = 0.75 m × 0.75 m × 0.75 m = 0.421875 cubic meters (m³)

This calculation gives us the total volume of the cube, which is the maximum amount of space available inside the vessel. Understanding the total volume is essential because it helps us to determine the proportion of the cube that is filled with water. This step provides a crucial reference point for our subsequent calculations. By knowing the vessel's total capacity, we can better understand the scale of the problem and the significance of the 225 liters of water poured into it. Accurate volume calculation is a key skill in geometry and has practical applications in various fields, such as construction and engineering.

Step 3: Determine the Height of the Water

Now that we know the volume of water in cubic meters (0.225 m³) and the side length of the cube (0.75 m), we can calculate the height to which the water rises. The volume of water in the cube forms a rectangular prism, and the volume of a rectangular prism is given by:

Volume = length × width × height

In this case, the length and width are both equal to the side length of the cube (0.75 m), and the volume is the volume of water (0.225 m³). We can rearrange the formula to solve for the height:

Height = Volume / (length × width)

Plugging in the values:

Height = 0.225 m³ / (0.75 m × 0.75 m) = 0.225 m³ / 0.5625 m² = 0.4 meters

Therefore, the height to which the water rises in the cubic vessel is 0.4 meters. This calculation directly answers the question posed in the problem. By understanding the relationship between volume, length, width, and height, we can accurately determine the water level inside the cube. This step demonstrates the practical application of geometric principles and highlights the importance of algebraic manipulation in problem-solving.

Final Answer

After carefully following the steps of unit conversion and volume calculation, we have determined that the water rises to a height of 0.4 meters in the cubic vessel. This corresponds to option c) in the original problem statement. To recap, we first converted the volume of water from liters to cubic meters, then calculated the total volume of the cubic vessel, and finally used the volume of water and the dimensions of the cube to find the height. This problem illustrates the importance of unit consistency and the application of geometric formulas in practical scenarios. The final answer provides a clear and concise solution to the problem, demonstrating the effectiveness of our step-by-step approach. This type of problem-solving skill is essential in various fields, including engineering, physics, and everyday applications involving volume and measurements.

Conclusion

In conclusion, solving the problem of finding the water height in a cubic vessel demonstrates a practical application of mathematical and geometric principles. We began by understanding the problem, which involved a cubic vessel with a given side length and a specific volume of water poured into it. The key to solving this problem was ensuring consistent units, which led us to convert liters to cubic meters. We then calculated the total volume of the cube and used the formula for the volume of a rectangular prism to determine the height of the water. The final answer, 0.4 meters, was obtained through careful calculation and application of mathematical formulas. This exercise underscores the importance of unit conversion, geometric formulas, and algebraic manipulation in solving real-world problems. The step-by-step approach used in this solution can be applied to various similar problems, making it a valuable skill for students and professionals alike. Understanding these concepts not only enhances mathematical proficiency but also provides a foundation for practical problem-solving in diverse fields.

In a cubic vessel with a side length of 0.75 m, 225 liters of water are poured. What is the height to which the water rises? The options are a) 0.2 m; b) 0.8 m; c) 0.4 m; d) 0.75 m.

Calculate Water Height in a Cubic Vessel A Step-by-Step Solution