Hexagonal Pitcher Volume And Orange Juice Calculation

by Scholario Team 54 views

Introduction

In this article, we will delve into a practical problem involving geometry and volume calculation. Specifically, we will determine the volume of a hexagonal pitcher and the amount of orange juice it holds when filled to 4/5 of its capacity. This exercise combines concepts of hexagonal prisms, area calculations, and volume determination, providing a comprehensive understanding of spatial reasoning and mathematical problem-solving. This topic falls under the domain of mathematics, particularly geometry and measurement. Understanding how to calculate the volume of three-dimensional shapes is crucial not only in academic settings but also in real-world applications such as engineering, architecture, and even everyday tasks like cooking and home improvement. To solve this problem effectively, we will need to recall the formula for the area of a regular hexagon and the formula for the volume of a prism. The area of a regular hexagon can be calculated using the formula: Area = (3√3 / 2) * side^2, where 'side' is the length of one side of the hexagon. Alternatively, if the apothem (the distance from the center to the midpoint of a side) is known, the area can be calculated as: Area = (Perimeter * Apothem) / 2. The volume of a prism is found by multiplying the area of its base by its height: Volume = Base Area * Height. The ability to apply these formulas correctly is essential for accurately determining the volume of the pitcher and the amount of orange juice it contains. This problem also involves a fractional part calculation, as we need to find 4/5 of the total volume to determine the amount of orange juice. Understanding fractions and how to apply them in real-world scenarios is another critical aspect of mathematical literacy. By working through this problem step-by-step, we will not only find the solution but also reinforce our understanding of these fundamental mathematical principles. So, let’s embark on this geometric journey and unlock the secrets of this hexagonal pitcher!

Problem Statement

We are given a hexagonal pitcher that is filled with orange juice up to 4/5 of its capacity. The hexagonal base has a side length of 3 cm, an apothem of 2.6 cm, and the pitcher's height is 30 cm. Our task is to determine two key values: the total volume of the pitcher and the amount of orange juice it contains. This problem requires us to apply our knowledge of geometry, specifically the properties of hexagons and prisms, as well as our understanding of volume calculations and fractions. To solve this, we will first calculate the area of the hexagonal base using the given side length or apothem. Once we have the base area, we can calculate the total volume of the pitcher by multiplying the base area by the height. Finally, we will determine the amount of orange juice by finding 4/5 of the total volume. This step involves multiplying the total volume by the fraction 4/5. By breaking down the problem into these smaller, manageable steps, we can systematically arrive at the solution. It's important to pay close attention to the units of measurement (centimeters in this case) and ensure that our calculations are accurate. This type of problem is not only a good exercise in mathematical skills but also demonstrates how geometry and measurement are used in practical situations. Whether it's determining the capacity of a container or calculating the amount of liquid it holds, these skills are valuable in various aspects of life. So, let's proceed with the calculations and find out the volume of the pitcher and the amount of orange juice it contains.

Calculating the Volume of the Hexagonal Pitcher

To begin, let's calculate the total volume of the hexagonal pitcher. Recall that the pitcher's base is a hexagon with a side length of 3 cm and an apothem of 2.6 cm, and the pitcher's height is 30 cm. The first step is to find the area of the hexagonal base. We can use the formula for the area of a regular hexagon when the apothem is known: Area = (Perimeter * Apothem) / 2. To find the perimeter of the hexagon, we multiply the side length by the number of sides, which is 6. So, the perimeter is 6 * 3 cm = 18 cm. Now we can plug the perimeter and apothem values into the area formula: Area = (18 cm * 2.6 cm) / 2. Calculating this gives us: Area = 46.8 cm² / 2 = 23.4 cm². Therefore, the area of the hexagonal base is 23.4 square centimeters. Next, we need to calculate the volume of the pitcher. The volume of a prism (which the pitcher is) is given by the formula: Volume = Base Area * Height. We have already calculated the base area as 23.4 cm², and we are given the height of the pitcher as 30 cm. Plugging these values into the volume formula, we get: Volume = 23.4 cm² * 30 cm. Calculating this product gives us: Volume = 702 cm³. So, the total volume of the hexagonal pitcher is 702 cubic centimeters. This value represents the maximum amount of liquid the pitcher can hold. It's important to note that the units for volume are cubic centimeters (cm³), as we are dealing with a three-dimensional space. This calculation demonstrates how the properties of geometric shapes, such as the area of a hexagon and the volume of a prism, can be used to solve practical problems involving capacity and measurement. Now that we know the total volume of the pitcher, we can move on to the next step: determining the amount of orange juice it contains when filled to 4/5 of its capacity.

Determining the Amount of Orange Juice

Now that we've successfully calculated the total volume of the hexagonal pitcher, which is 702 cm³, our next step is to determine how much orange juice the pitcher contains when it's filled to 4/5 of its capacity. This involves a simple fraction calculation. We need to find 4/5 of the total volume. To do this, we multiply the total volume by the fraction 4/5: Orange Juice Volume = (4/5) * Total Volume. Plugging in the total volume we calculated earlier, we get: Orange Juice Volume = (4/5) * 702 cm³. To perform this calculation, we can first multiply 702 by 4, which gives us 2808: Orange Juice Volume = 2808 cm³ / 5. Now, we divide 2808 by 5 to get the final volume of orange juice: Orange Juice Volume = 561.6 cm³. Therefore, the amount of orange juice in the pitcher when it's filled to 4/5 of its capacity is 561.6 cubic centimeters. This result gives us a clear understanding of the pitcher's capacity in a practical context. It shows how much liquid is actually present in the pitcher, as opposed to its maximum possible volume. This type of calculation is common in everyday situations, such as cooking, where we often need to measure ingredients as fractions of a whole. It also highlights the importance of understanding fractions and how to apply them in real-world scenarios. In summary, we have successfully determined that the hexagonal pitcher contains 561.6 cm³ of orange juice when filled to 4/5 of its capacity. This completes our solution to the problem.

Conclusion

In conclusion, we have successfully solved the problem of determining the volume of a hexagonal pitcher and the amount of orange juice it contains when filled to 4/5 of its capacity. We found that the total volume of the pitcher is 702 cm³ and the amount of orange juice it holds is 561.6 cm³. This problem involved several key steps, including calculating the area of a hexagon, determining the volume of a prism, and applying fractions to find a portion of the total volume. Each of these steps required a solid understanding of geometric principles and mathematical operations. By breaking down the problem into smaller, manageable parts, we were able to systematically arrive at the solution. This approach is a valuable strategy for tackling complex problems in mathematics and other fields. The skills we used in this exercise, such as calculating areas and volumes, are fundamental in various real-world applications. From engineering and architecture to cooking and everyday measurements, the ability to work with geometric shapes and quantities is essential. Furthermore, the problem highlighted the importance of understanding fractions and how to apply them in practical contexts. Whether it's determining the quantity of an ingredient or calculating a percentage, fractions play a crucial role in many aspects of life. This exercise also served as a reminder of the interconnectedness of mathematical concepts. Geometry, measurement, and arithmetic are not isolated topics but rather pieces of a larger puzzle. By mastering these fundamental concepts, we can develop a strong foundation for further mathematical studies and problem-solving. In summary, this problem provided a comprehensive review of essential mathematical skills and demonstrated their practical relevance. We hope that this detailed explanation has enhanced your understanding of volume calculations and the application of mathematical principles in real-world scenarios.