Calculating the Volume of the Displaced Water

Oct 13, 2025 by wbrum.org 97 views

Introduction

In various fields such as engineering, architecture, and geology, measuring the volume of irregularly shaped objects is a crucial task. One common method used to determine the volume of such objects is by using a container with a known volume and measuring the displacement of water when the object is submerged. In this article, we will discuss how to measure the volume of an irregularly shaped stone using a cylindrical container and the principle of displacement.

The Problem

Ana, a student, was tasked with measuring the volume of an irregularly shaped stone. She used a cylindrical container with a radius of 8 cm and filled it with water up to a height of 20 cm. After placing the stone in the container, the height of the water rose to 23.5 cm. The problem is to find the volume of the stone.

The Solution

To solve this problem, we need to understand the concept of displacement and how it relates to the volume of the stone. When the stone is submerged in the water, it displaces a volume of water equal to its own volume. The volume of the displaced water is equal to the difference in the volume of water in the container before and after the stone is submerged.

Calculating the Volume of the Displaced Water

The volume of the displaced water can be calculated using the formula for the volume of a cylinder:

V = πr^2h

where V is the volume, π is a constant approximately equal to 3.14, r is the radius of the cylinder, and h is the height of the water.

In this case, the radius of the cylinder is 8 cm, and the initial height of the water is 20 cm. After the stone is submerged, the height of the water rises to 23.5 cm. Therefore, the volume of the displaced water is:

V = π(8)^2(23.5 - 20) = π(64)(3.5) = 3.14(64)(3.5) = 703.36 cm^3

Calculating the Volume of the Stone

The volume of the stone is equal to the volume of the displaced water. Therefore, the volume of the stone is:

V = 703.36 cm^3

Conclusion

In this article, we discussed how to measure the volume of an irregularly shaped stone using a cylindrical container and the principle of displacement. We calculated the volume of the displaced water and used it to determine the volume of the stone. The volume of the stone is 703.36 cm^3.

Mathematical Formulas

V = πr^2h
V = π(8)^2(23.5 - 20)
V = π(64)(3.5)
V = 3.14(64)(3.5)
V = 703.36 cm^3

References

FUVEST 2023
Mathematics textbooks and online resources

Future Work

This problem can be extended to more complex shapes and objects. Future work can involve developing more accurate methods for measuring the volume of irregularly shaped objects and applying these methods to real-world problems.

Limitations

This method assumes that the stone is fully submerged in the water and that the container is cylindrical. In reality, the stone may not be fully submerged, and the container may not be cylindrical. Future work can involve developing methods to account for these limitations.

Conclusion

Q: What is the principle behind measuring the volume of an irregularly shaped stone?

A: The principle behind measuring the volume of an irregularly shaped stone is based on the concept of displacement. When the stone is submerged in a container filled with water, it displaces a volume of water equal to its own volume. By measuring the volume of the displaced water, we can determine the volume of the stone.

Q: What type of container is used to measure the volume of an irregularly shaped stone?

A: A cylindrical container is typically used to measure the volume of an irregularly shaped stone. The container is filled with water to a certain height, and the stone is then submerged in the water. The height of the water after the stone is submerged is measured, and the volume of the displaced water is calculated.

Q: How is the volume of the displaced water calculated?

A: The volume of the displaced water is calculated using the formula for the volume of a cylinder:

V = πr^2h

where V is the volume, π is a constant approximately equal to 3.14, r is the radius of the cylinder, and h is the height of the water.

Q: What are the limitations of this method?

A: This method assumes that the stone is fully submerged in the water and that the container is cylindrical. In reality, the stone may not be fully submerged, and the container may not be cylindrical. Future work can involve developing methods to account for these limitations.

Q: Can this method be used to measure the volume of other objects?

A: Yes, this method can be used to measure the volume of other objects, such as irregularly shaped rocks, soil, or other materials. The principle of displacement is a fundamental concept in physics and can be applied to a wide range of objects.

Q: How accurate is this method?

A: The accuracy of this method depends on the precision of the measurements and the assumptions made about the object being measured. In general, this method can provide accurate results for objects with simple shapes, but may not be as accurate for objects with complex shapes or irregular surfaces.

Q: What are some real-world applications of this method?

A: This method has a wide range of real-world applications, including:

Geology: Measuring the volume of rocks and soil to understand geological processes and formations.
Engineering: Measuring the volume of materials to design and construct buildings, bridges, and other structures.
Environmental science: Measuring the volume of water and other fluids to understand and manage environmental systems.

Q: Can this method be used to measure the volume of objects in other fluids?

A: Yes, this method can be used to measure the volume of objects in other fluids, such as oil, gas, or other liquids. The principle of displacement is a fundamental concept in physics and can be applied to a wide range of fluids.

Q: What are some potential future developments in this area?

A: Some potential future developments in this area include:

Developing more accurate methods for measuring the volume of irregularly shaped objects.
Applying this method to more complex shapes and objects.
Developing methods to account for limitations and assumptions made about the object being measured.

Conclusion

In conclusion, measuring the volume of an irregularly shaped stone using a cylindrical container and the principle of displacement is a useful technique in various fields. By understanding the concept of displacement and using mathematical formulas, we can accurately determine the volume of such objects. This technique can be extended to more complex shapes and objects, and future work can involve developing more accurate methods for measuring the volume of irregularly shaped objects.