> For the complete documentation index, see [llms.txt](https://wenbo-notes.gitbook.io/coding/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://wenbo-notes.gitbook.io/coding/kattis/easy/beavergnaw.md).

# Beavergnaw

## Question

{% embed url="<https://open.kattis.com/problems/beavergnaw>" %}

## Solution

### Idea

This is purely a math problem, a math problem about the solid geometry. Quite boring since it took me a while to derive the math formula. The steps are summarised as follows:

1. Calculate the volumne of the bigger cylinder `total`, which is $$\pi\*(\frac{D}{2})^2\*D$$
2. Use `total` minus the input `V` to get the volume of the wood eaten by the beaver, $$\text{wood}=\text{total}-\text{v}$$

Now the trickiest thing begins, we need to deduct the two truncated cone. Here we use the $$\text{bigCone}-\text{smallCone}$$ to calculate. And we first consider the upper half.

Notice that for the upper half, the $$\text{bigCone}$$ is with both radius and height as $$\frac{D}{2}$$, while the $$\text{smallCone}$$ is with both radius and height as $$\frac{d}{2}$$. So, the volume for the upper truncated cone should be $$\frac{1}{3}(\pi(\frac{D}{2})^3-\pi(\frac{d}{2})^3)$$.

So, for the two truncated cones, their total volume should be $$\frac{2}{3}(\pi(\frac{D}{2})^3-\pi(\frac{d}{2})^3)$$, which is same as $$\frac{1}{3}(\pi(\frac{D}{2})^2\cdot D-\pi(\frac{d}{2})^2\cdot d)$$.

Notice that the volumne of two truncated cones plus the volume of the inner cylinder will be the volumne of $$\text{wood}$$. And the volumne of the cylinder is given by $$\pi(\frac{d}{2})^2\cdot d$$, which can be simplify to $$2\pi r^3,\text{where }r=\frac{d}{2}$$. (Now, you should be able to solve for the $$d$$). Manipulate the formula, we have $$\frac{2}{3}(2\pi r^3)=\text{wood}-\frac{1}{3}\pi(\frac{D}{2}^2)\cdot D$$

3. Now, let's define a new variable `bigCones`, whose value is $$\frac{1}{3}\pi(\frac{D}{2})^2\cdot D$$
4. Then, define another intermediate variable `shape`, whose value is $$\text{wood}-\text{bigCones}$$
5. Last and finally, define the last variable `cylinder` to represent the volume of the inner cylinder, whose value is $$\text{shape}\cdot \frac{3}{2}$$.&#x20;
6. Now, our $$d=2\cdot r=2\cdot\sqrt\[3]{\frac{\text{cylinder}}{2\pi}}$$

### Code

{% @github-files/github-code-block url="<https://github.com/mendax1234/Coding-Problems/blob/main/kattis/beavergnaw/beavergnaw.c>" %}


---

# Agent Instructions
This documentation is published with GitBook. GitBook is the documentation platform designed so that both humans and AI agents can read, navigate, and reason over technical content effectively. Learn more at gitbook.com.

## Querying This Documentation
If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.

Perform an HTTP GET request on the current page URL with the `ask` query parameter:

```
GET https://wenbo-notes.gitbook.io/coding/kattis/easy/beavergnaw.md?ask=<question>
```

The question should be specific, self-contained, and written in natural language.
The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.