# Determine the Volume of Space the Astronaut Can Visit Around the Cube

An astronaut is tethered to a vertex of a cube with a side length of $$3$$ meters. What is the total volume of the space (in cubic meters) that he is able to visit if the length of the rope is $$4$$ meters?

Bonus: What if the rope length is $$3(1 + \sqrt{2})$$ meters?

The first part of the problem was originally asked here.

The second part of the problem is very interesting since the rope now extends to the opposite vertex. It is clear that the minimum distance between two opposing vertices is $$3\sqrt{5}$$, so the remaining length is $$3(1 + \sqrt{2} - \sqrt{5})$$. But does that necessarily mean the astronaut can go that far around the opposing vertex?

Note by Michael Huang
9 months, 1 week ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$