Answer
$1$ft
Work Step by Step
Recall that the formula for the surface area of a cylinder is:
$$(2\pi{r}^2)+(2\pi{r}h) = S$$
where $(2\pi{r}^2)$ is the area of the top and the bottom circles,
and $(2\pi{r}h)$ is the area of a rectangle
with the circumference of the circle, $2\pi{r}$, as one side
and the height of the cylinder, $h$, as the other
If the cylinder is $3$ft high and the surface area is $8\pi$ft$^2$ we can plug this into our formula:
$(2\pi{r}^2)+(2\pi{r}3)=8\pi$
$2\pi{r}^2+6\pi{r}=8\pi$
We can rewrite in standard quadratic equation form: $ar^2 + br +c=0$
$2\pi{r}^2 +6\pi{r}-8\pi=0$
where $a=2\pi$, $b=6\pi$, and $c=-8\pi$
Now apply the quadratic formula: $r=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$
$r=\dfrac{(-)(6\pi)\pm\sqrt{(6\pi)^2-4(2\pi)(-8\pi)}}{2(2\pi)}$
$r=\dfrac{-6\pi\pm\sqrt{36\pi^2-4(-16\pi^2)}}{4\pi}$
$r=\dfrac{-6\pi\pm\sqrt{36\pi^2+64\pi^2}}{4\pi}$
$r=\dfrac{-6\pi\pm\sqrt{100\pi^2}}{4\pi}$
$r=\dfrac{-6\pi\pm10\pi}{4\pi}$
$r=\dfrac{-6\pi+10\pi}{4\pi}$ or $r=\dfrac{-6\pi-10\pi}{4\pi}$
$r=\dfrac{4\pi}{4\pi}$ or $r=\dfrac{-16\pi}{4\pi}$
$r=1$ or $r=-4$
Since the length of the radius has to be positive, $r=1$ft
