Calculated Rise in Water Level: A Cone Submerged in a Cylindrical Jar

In this article, we will explore the process of calculating the rise in water level when a solid cone is submerged in a cylindrical jar. This problem involves the principles of volume calculations, specifically the volume of a cone and the displacement method. We will break down the steps and provide detailed explanations, along with calculations to understand the rise in water level.

Introduction to the Problem

The problem involves a solid cone with a height of 4 cm and a base radius of 2 cm. This cone is submerged into a cylindrical jar with a radius of 6 cm, which is filled with water. The goal is to determine how much the water level rises as a result of the cone being submerged.

Calculating the Volume of the Cone

The volume of a cone is given by the formula:

V_{text{cone}} frac{1}{3} pi r^2 h

Where:

- V_{text{cone}} is the volume of the cone. - pi is a mathematical constant approximately equal to 3.14159. - r is the radius of the base of the cone. - h is the height of the cone.

Given Values

- Height of the cone: h 4 cm - Radius of the cone: r 2 cm

Substituting the Values

V_{text{cone}} frac{1}{3} pi 2^2 4 frac{1}{3} pi 16 frac{16}{3} pi text{ cm}^3

Calculating the Rise in Water Level

When the cone is submerged, the volume of water displaced is equal to the volume of the cone. This displaced volume will cause the water level in the cylindrical jar to rise. The rise in water level can be calculated using the volume of a cylinder formula:

V_{text{cylinder}} pi R^2 h_{text{rise}}

Where:

- V_{text{cylinder}} is the volume of the cylinder. - pi is a mathematical constant approximately equal to 3.14159. - R is the radius of the cylindrical jar. - h_{text{rise}} is the rise in water level.

Given Values

- Radius of the jar: R 6 cm - Volume displaced (equal to the volume of the cone): V_{text{cylinder}} frac{16}{3} pi text{ cm}^3

Setting the Volumes Equal

frac{16}{3} pi pi 6^2 h_{text{rise}}

Simplifying the Equation

frac{16}{3} 36 h_{text{rise}} h_{text{rise}} frac{16}{3 times 36} frac{16}{108} frac{4}{27} text{ cm}

The rise in water level when the cone is submerged in the cylindrical jar is:

boxed{frac{4}{27} text{ cm}}

Further Exploration

The volume of the cone is also solved with the derived formula:

V_{text{cone}} 1/3 pi r^2 h 1/3 pi 2^2 4 16.755 text{ cm}^3

The volume of the cylinder is solving using the formula:

V_{text{cylinder}} pi r^2 h pi 6^2 X

Where X is the height of the displaced water. Equating the volume of the displaced water to the volume of the solid cone:

pi 6^2 X 16.755 text{ cm}^3

Solving for the Height of the Displaced Water

36X 16.755

X 16.755 / 36 pi 0.148 text{ cm}

The rise in water due to the cone is:

0.148 text{ cm}

Conclusion

In conclusion, the rise in water level when a solid cone with a height of 4 cm and a base radius of 2 cm is submerged in a cylindrical jar with a radius of 6 cm is equal to 4/27 cm. This problem demonstrates the application of volume calculations and the principle of water displacement in determining the rise in water level.

Frequently Asked Questions (FAQ)

Q: How can I verify the calculations?

A: You can verify the calculations by plugging in the given values into the formulas and performing the arithmetic step-by-step. Ensure that the units are consistent and that you use the correct values for pi.

Q: What other real-life applications use the principle of water displacement?

A: The principle of water displacement has numerous real-life applications, including determining the density of substances, measuring the volume of irregularly shaped objects, and in engineering and environmental science.

Q: Are there any common errors in calculating the rise in water level?

A: Common errors include incorrect substitution of values, misapplication of formulas, and units mix-up. Always double-check your work and ensure that all calculations are performed with the correct values and units.