I recently encountered a question from my math textbook (I write as a Grade 12 math student) which presented a radical equation to be solved. I did so, but in a messy way. Writing this question serves me the purpose of having it simplified.
The equation is simple: $160=\pi r \sqrt{36 + r^2}$.
My approach is:
\begin{align} 160 &=\pi r\sqrt{36 + r^2}\\ 25600 &=\pi^2 r^2 (36 + r^2)\\ 0 &= r^4 \pi^2 + 36r^2 \pi^2 - 25600\\ 0 &= \pi^2(r^4 + 36r^2) - 25600\\ 0 &= \pi^2(r^4 + 36r^2 + 18^2) - 18^2 \pi^2 - 25600\\ 0 &= \pi^2(r^2 + 18)^2 - 324\pi - 25600\\ \sqrt{\frac{25600 + 324\pi}{\pi^2}} &= r^2 + 18\\ r &= \sqrt{\sqrt{\frac{25600 + 324\pi}{\pi^2}}-18} \approx 6\\ \end{align}
The textbook gave an exact answer of 6. This prompted me to think there is a neater and "prettier" way of solving this equation. Please help.