Ok. I have a Surface Area Formula for a cylinder, but I am wanting to solve for the radius. The only information I have is the height of the cylinder, which is 8 inches. I know I can use the Quadratic Formula to convert the formula to solve for the radius, but I get stuck when doing the math.
$$ SA=2\pi r^{2}+2\pi rh$$
Any help?
On
I'd just complete the square:
$$ A = 2\pi r^2 + 2\pi rh $$
$$ \frac{A}{2\pi} = r^2 + rh $$
To complete the square on the right-hand side, we need to add $\frac{h^2}{4} $ to both sides:
$$ \frac{A}{2\pi} + \frac{h^2}{4} = r^2 + rh + \frac{h^2}{4} $$
$$ \frac{A}{2\pi} + \frac{h^2}{4} = \left(r + \frac{h}{2} \right)^2 $$
Since $r > 0$ and $h > 0$, we take the positive root:
$$ r + \frac{h}{2} = \sqrt{\frac{A}{2\pi} + \frac{h^2}{4}} $$
$$ r = -\frac{h}{2} +\sqrt{\frac{A}{2\pi} + \frac{h^2}{4}} $$
If you have
$$A=2\pi r^{2}+2\pi rh$$
then rearrange it:
$$2\pi r^{2}+2\pi hr - A = 0$$
This is quadratic in $r$ with solutions:
$$r = \frac{-2\pi h \pm \sqrt{4\pi^2h^2 + 8\pi A}}{4\pi}.$$
Since the expression under the radical is greater than $2\pi h$ you choose the "plus" root so that the radius is positive:
$$r = \frac{-2\pi h + \sqrt{4\pi^2h^2 + 8\pi A}}{4\pi}.$$
But to solve for the numerical value of $r$ given $h=8$ inches, you'll need more information. You'll need the surface area also to get the value of $r$.