Calculating the Volume of a Floating Sphere Partly Submerged in Water

Imagine a solid sphere with a diameter of 10 cm floating in water such that the top 2 cm of the sphere is above the water surface. This scenario raises an interesting question: how much of the sphere is submerged? In this article, we will explore the steps to calculate the volume of the submerged part of the sphere.

Understanding the Problem

The core of the problem is determining the volume of the submerged portion of the sphere. Since the top of the sphere is 2 cm above the water surface, the height of the submerged part (cap) can be calculated.

Step-by-Step Calculation

Determine the Radius of the Sphere

The diameter of the sphere is 10 cm. Therefore, the radius r is:

[ r frac{text{diameter}}{2} frac{10 text{ cm}}{2} 5 text{ cm} ]

Calculate the Height of the Submerged Part

The height h of the submerged part of the sphere can be determined by subtracting the height above the water surface from the total diameter:

[ h text{diameter} - text{height above water} 10 text{ cm} - 2 text{ cm} 8 text{ cm} ]

Use the Formula for the Volume of a Spherical Cap

The volume V of the submerged part of the sphere, which is a spherical cap, can be calculated using the formula:

[ V frac{1}{3} pi h^2 left(3r - hright) ]

Plugging in the values:

[ V frac{1}{3} pi left(8 text{ cm}right)^2 left(3 times 5 text{ cm} - 8 text{ cm}right) ] [ V frac{1}{3} pi left(64 text{ cm}^2right) left(15 text{ cm} - 8 text{ cm}right) ] [ V frac{1}{3} pi left(64 text{ cm}^2right) left(7 text{ cm}right) ] [ V frac{1}{3} pi left(448 text{ cm}^3right) ] [ V frac{448}{3} pi text{ cm}^3 ]

Calculate the Numerical Value

Using pi approx 3.14, the numerical value of the submerged volume is:

[ V approx frac{448}{3} times 3.14 text{ cm}^3 approx 469.33 text{ cm}^3 ]

Therefore, the volume of the part of the sphere that is submerged below the water is approximately 469.33 cm3.

Alternative Methods

One could look up the formula for the volume of a spherical cap, the top 2 cm of the sphere, and the volume of a sphere. The difference between these volumes would give the submerged volume. Alternatively, the method of horizontal slices and calculus can be used for a more detailed approach, though this is already known to be simpler. Another shortcut involves using the volume of a sphere and subtracting the volume of the top 2 cm spherical cap.

For your reference:

The volume of the sphere can be calculated as:

[ V frac{4}{3} pi r^3 ]

The volume of the top 2 cm spherical cap can be derived using integration or known formulas.

For a more practical approach, one could measure the volume of water displaced when the sphere is submerged by a certain amount.