Question

ut%20out%20exactly%20%7D" id="TexFormula1" title="\text{I was eating cookies and had some thoughts. If I wanted to cut out exactly }" alt="\text{I was eating cookies and had some thoughts. If I wanted to cut out exactly }" align="absmiddle" class="latex-formula">$\frac{1}{3}$of the cookie to share with someone, how far from one side would I have to make a straight cut to get that exact amount? How far would I have to cut if I wanted to cut off$\frac{1}{n}$$\text{ of the cookie?}$
$\text{Basically, the question is, find the value of }a\text{ given only n, and r}$

$\text{One way of finding this, is by finding the area of the shaded reigon, Q in terms of}$
$\text{r, a, and b, and equating it to the area of the fraction of the cookie then solving for a.}$

$\text{In math, this means solving } \frac{1}{n}\pi r^2=Q \text{ for }f(r,n)=a.$

$\text{From the diagram, we can see that }r=a+b$

$\text{Eventually, by 2 different means, I found 2 equations that, if solved, would give the}$$\text{ relationship between r, n, and a.}$$\text{They are as follows:}$

$\text{1. }\frac{1}{n}\pi r=r\theta-bsin(\theta) \text{ where }\theta=cos^{-1}(\frac{b}{r})$

$\text{2. }\frac{1}{n}\pi=\theta-sin(2\theta)\text{ where }\theta=cos^{-1}(\frac{b}{r})$

$\text{These 2 equations are equivalent, but annoying to solve.}$

$\text{To claim these points, please solve for a in terms of r and n, showing all work.}$
$\text{I would like an analytic solution if possible.}[\tex][tex]\text{All incorrect, spam, or no-work solutions will be reported.}$

Answer 1

In the attachement, there is what I came up with so far. I think that finding 'a' is non-trivial, if possible at all.

$A_c$ - the area of a circle

$A_{cs}$ - the area of a circular segment