Is there an algebraic solution to this problem? The base of my pool cartridge filter tank is composed of an arc and a chord and has a total perimeter of 49.5 inches.  The length of the chord portion of the shape is 7 inches.  The arc portion has a length of 42.5 inches.  Using this information I wanted to find the length of the radius of the arc.
I couldn’t do it algebraically.  I had to come up with an approximate solution numerically.  The equation that I formulated ended up being a transcendental equation.  When attempting to solve, I got stuck when the equation was roughly equivalent to r=sin(r).
Is there an algebraic solution to this problem?  If so, I would love to see what it is and how you got there.  
 
 A: I agree: when I write down the relevant equations, I find that $r$ satisfies
$$
\frac{3.5}r = \sin \frac{42.5}{2r}.
$$
Such an equation has no elementary solution. Its approximate solution is $7.918068\dots$, which of course is far more precise than any measurements one would make.
A: We all agree on the fact that the equation is transcendental. But, it is quite easy to solve.
Starting from what Greg Martin wrote, let us define $x=\frac{42.5}{2r}$ and the equation then write $$\frac{14 }{85}x=\sin(x)$$ Now, replace the sine by this beautiful approximation proposed more than 1400 years ago  by Bhaskara I (c. 600 – c. 680), a seventh-century Indian mathematician and astronomer $$\sin(x)\approx \frac{16x(\pi-x)}{5\pi^2-4x(\pi-x)}$$ Using it, we face now a quadratic equation $$28 x^2-(28 \pi -680) x+(35 \pi -680) \pi=0$$ the positive solution of which being $$x=\frac{1}{56} \left(28 \pi -680+\sqrt{462400+38080 \pi -3136 \pi ^2}\right)\approx 2.68416$$ from which $r\approx 7.91683$.
There is another way to approach the solution, building a Pade approximant at $x=0$; a simple one would be $$\sin(x)=\frac{\frac{551 x^5}{166320}-\frac{53 x^3}{396}+x}{\frac{5 x^4}{11088}+\frac{13
   x^2}{396}+1}$$ which will lead to a quadratic equation in $x^2$ $$\frac{9157 x^4}{33264}-\frac{4687 x^2}{396}+71=0$$ the solution of which being $$x=\sqrt{\frac{6 \left(32809-\sqrt{475694653}\right)}{9157}}\approx 2.68452$$ from which $r\approx 7.91574$.
Using instead a $[7,7]$ Pade approximant for $\sin(x)$ built at $x=0$ will lead to a cubic equation in $x^2$ the solution of which being $x\approx 2.68373$ from which $r\approx 7.91807$ which is almost identical to Greg Martin's result.
