Question

Apply the formulas for volume of a sphere and a cube.
|
Identify perfect cubes.
Understand _.
Determine side length and _ measures.

Answer 1

To understand the volume of a sphere and a cube, as well as to identify perfect cubes, let's break this down step by step.

### 1. Volume Formulas
- **Volume of a Cube**: The formula for the volume \( V \) of a cube with side length \( s \) is given by:
\[
V = s^3
\]
- **Volume of a Sphere**: The formula for the volume \( V \) of a sphere with radius \( r \) is given by:
\[
V = \frac{4}{3} \pi r^3
\]

### 2. Identifying Perfect Cubes
A perfect cube is a number that can be expressed as \( n^3 \) where \( n \) is an integer. Examples of perfect cubes include:
- \( 1^3 = 1 \)
- \( 2^3 = 8 \)
- \( 3^3 = 27 \)
- \( 4^3 = 64 \)
- \( 5^3 = 125 \)
- \( 6^3 = 216 \)
- \( 7^3 = 343 \)
- \( 8^3 = 512 \)
- \( 9^3 = 729 \)
- \( 10^3 = 1000 \)

### 3. Understanding Side Length and Volume Measures
To find the side length of a cube given its volume, you can rearrange the volume formula:
\[
s = \sqrt[3]{V}
\]
And for a sphere, if you know its volume and you want to find the radius, you can rearrange the volume formula:
\[
r = \sqrt[3]{\frac{3V}{4\pi}}
\]

### Summary
- **Volume of a Cube**: \( V = s^3 \)
- **Volume of a Sphere**: \( V = \frac{4}{3} \pi r^3 \)
- **Perfect Cubes**: Numbers like 1, 8, 27 are examples.
- **Side Length Calculation**: For a cube, \( s = \sqrt[3]{V} \) and for a sphere, \( r = \sqrt[3]{\frac{3V}{4\pi}} \).

This structure allows you to understand the relationships between volumes, perfect cubes, and their properties.