Question

What is the ratio of the volume of a cube of side \(r\) to that of a sphere of radius \(r\) ? Does your answer depend on the particular value of \(r ?\)

Short answer

Answer: The ratio of the volume of a cube to the volume of a sphere when both have the same length r is \(\frac{3}{4\pi}\). This ratio does not depend on the particular value of r.

Step-by-step solution

Find the volume of the cube

First, we need to find the formula for the volume of a cube with side length \(r\). The volume of a cube is given by: Volume of cube = Side × Side × Side So, the volume of our cube with side length \(r\) is: \(V_{cube} = r^3\)

Find the volume of the sphere

Next, we need to find the formula for the volume of a sphere with radius \(r\). The volume of a sphere is given by: Volume of sphere = \(\frac{4}{3}\)π × Radius³ So, the volume of our sphere with radius \(r\) is: \(V_{sphere} = \frac{4}{3}\pi r^3\)

Find the ratio between the volumes of the cube and the sphere

Now, we have the volumes of both shapes: \(V_{cube} = r^3\) and \(V_{sphere} = \frac{4}{3}\pi r^3\). To find the ratio of their volumes, we will divide the volume of the cube by the volume of the sphere: Ratio = \(\frac{V_{cube}}{V_{sphere}}\) Plug in the formulas for the volumes: Ratio = \(\frac{r^3}{\frac{4}{3}\pi r^3}\)

Simplify the ratio expression

To simplify the ratio, we can cancel the \(r^3\) term in the numerator and the denominator: Ratio = \(\frac{r^3}{\frac{4}{3}\pi r^3} = \frac{1}{\frac{4}{3}\pi}\) Now, we can further simplify the expression: Ratio = \(\frac{3}{4\pi}\)

Check for dependence on \(r\)

Looking at our final ratio expression, \(\frac{3}{4\pi}\), we see that it does not contain the variable \(r\). This means that the ratio of the volume of a cube to the volume of a sphere with side length or radius \(r\) does not depend on the particular value of \(r\).

Key concepts

Cube Volume

A cube is a three-dimensional shape with equal sides, making it simple and elegant. To find the volume of any cube, you just need the length of its side. If a cube has a side length labeled as \( r \), then calculating its volume is straightforward. Simply use the formula:

• Volume of a cube = Side × Side × Side

For our cube, this becomes:

• \( V_{cube} = r^3 \)

This formula is simply multiplying the side length \( r \) three times since a cube has the same dimension along all three axes. By understanding this formula, you can easily calculate the volume for any size of a cube.
Remember, the unit of volume will always be cubic units, matching the unit used for the side length.

Sphere Volume

Spheres are fascinating shapes with every point on their surface equidistant from the center, known as the radius. To determine the volume of a sphere, you'll use its radius \( r \). The formula for the volume of a sphere is a bit more complex due to the involvement of the constant \( \pi \), which is approximately 3.14159. Here's the formula:

• Volume of sphere = \( \frac{4}{3}\pi r^3 \)

In this equation, you multiply the cubed radius by \( \pi \) and then by \( \frac{4}{3} \). This reflects how a sphere's volume scales with its radius. So, the unit of volume here is also in cubic units, corresponding to the input's unit for the radius.
The appearance of \( \pi \) shows the connection of a sphere with circular geometry, which is a hallmark of its structure.

Mathematical Ratios

A ratio is a way to compare two quantities by showing how many times one value contains or is contained within the other. When dealing with volumes, a ratio helps us understand how different space-filling shapes compare. In the given exercise, the goal is to find the ratio between volumes of a cube and a sphere, both with characteristic length \( r \).To find the ratio, you take the volume of the cube and divide it by the volume of the sphere:

• \( \text{Ratio} = \frac{V_{cube}}{V_{sphere}} \)

Plugging in the formulas for each volume, this becomes:

• \( \text{Ratio} = \frac{r^3}{\frac{4}{3}\pi r^3} \)

Upon simplifying, the \( r^3 \) terms cancel each other out, leading to:

• \( \text{Ratio} = \frac{3}{4\pi} \)

This ratio, \( \frac{3}{4\pi} \), intriguingly does not depend on the size of \( r \), meaning the way these shapes' volumes compare remains constant, regardless of their absolute size.
This reflects an important property of mathematical ratios - they often reveal insights and properties that are consistent across changes in scale.