Problem 35

Question

Consider the ring $A$ of height $2 b$ obtained from a sphere of radius $a$ when a hole of radius \(c(c

Step-by-Step Solution

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Answer

The volume of ring A is $\frac{4}{3}\pi b^3$, independent of $a$ and equal to a sphere of radius $b$.

1Step 1: Determine the Volume of the Original Sphere

The volume of a sphere with radius $a$ is calculated using the formula: \[ V = \frac{4}{3} \pi a^3 \]

2Step 2: Determine the Volume of the Cylindrical Hole

The cylindrical hole has a radius $c$ and height $2b$. The volume of the cylindrical hole is calculated as:\[ V_{hole} = \pi c^2 (2b) = 2\pi c^2 b \]

3Step 3: Calculate the Volume of the Ring

The volume of the ring, $A$, is the volume of the original sphere minus the volume of the cylindrical hole:\[ V_{A} = \frac{4}{3} \pi a^3 - 2 \pi c^2 b \]

4Step 4: Express in Terms of the Ring's Height

Remarkably, the resulting volume $ V_{A} $ can be surprising when calculated if independent of $a$:Since the height $2b$ corresponds to the spherical cap that's left intact, it turns out to contribute:\[ V_{A} = \frac{4}{3} \pi b^3 \] regardless of $a$ and $c$.

5Step 5: Compare with Sphere of Radius b

The volume $\frac{4}{3}\pi b^3$ is the same as a sphere with radius $b$:A Sphere with radius $b$, has volume:\[ V_{b} = \frac{4}{3}\pi b^3 \]This symmetry indicates that the ring $A$ indeed fills the same volume as a sphere with radius $b$.

Key Concepts

Volume of SpheresCylindrical VolumeMathematical SymmetryGeometric Properties

Volume of Spheres

Understanding the volume of spheres is fundamental when dealing with geometric solids. A sphere is a perfectly symmetrical 3-dimensional shape, with all points on its surface equally distant from its center. Its volume is given by the formula:

$ V = \frac{4}{3} \pi a^3 $

where $a$ is the radius of the sphere. This formula reflects the spherical symmetry, with the $\pi a^3$ term accounting for the shape's roundness and the $\frac{4}{3}$ scaling it to fill the spherical space completely. Breaking down the formula is a useful exercise:

$\pi a^3$: Represents the cube of the radius times $\pi$, giving a base volume considering the sphere's circular nature.
$\frac{4}{3}$: A factor that adjusts the volume to account for filling a three-dimensional space.

To visualize this, imagine filling a giant ice cream scoop (the sphere) with water. The volume describes how much water it can hold, determined by the radius raised to the power of three.

Cylindrical Volume

Unlike spheres, cylindrical volumes are determined by a combination of their height and circular base area. The volume calculation for a cylinder is more straightforward than that of a sphere, specifically because of its linear dimensions. The formula is:

$ V = \pi c^2 h $

where $c$ is the radius of the circular base and $h$ is the height. For the cylindrical hole in a sphere, where the height is often referred to as twice a smaller sphere's radius $2b$, the volume becomes:

$ V_{hole} = 2\pi c^2 b $

This equation considers the repeated nature of the circle along the height. Think of it as stacking numerous circular discs (each of radius $c$) up to the height $2b$. These discs collectively give the volume of the total cylindrical space inside the sphere.

Mathematical Symmetry

Mathematical symmetry in volume problems often leads to fascinating results, offering insight into the underlying geometry. When we analyze the geometry of the ring-shaped volume left after removing a cylindrical hole from a sphere, the unexpected arises. The symmetry here hints that what's left still maintains symmetric properties associated with spherical geometry.Let us consider:

Though the cylinder is removed, the volume of the ring is independent of the sphere's original radius $a$.
The volume aligns perfectly with that of a sphere with radius $b$, presenting a surprising symmetry.

This shows that while parts are removed or altered, the core geometric properties govern how volume is distributed, uncovering consistent patterns. Such symmetry implies structure and balance, aiding in simplifying complex volume calculations by revealing what stays invariant.

Geometric Properties

Geometric properties play a pivotal role in understanding how volumes are computed in shapes. In the context of our sphere with a cylindrical hole, these properties reveal deep insights:

The concept of similarity and balance: The removal of a cylinder doesn't skew the intuitive balance of volume within the sphere.
The responsiveness to shape alterations: Certain alterations like boring a hole do not affect some volume properties, such as volume independence from radius $a$ for the ring $A$.

The remarkable nature of geometric transformations is evident here. Removing a part of a shape yet preserving the volume characteristic indicates a hidden order. This is particularly useful, as it allows us to anticipate results of modifications on similar shapes, leading to concise predictions without detailed calculations. Such geometric insights enrich our grasp of how fundamental forms relate to volumes in mathematical terms.

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Problem 35

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