Find the radius of circle with a regular octagon and a dodecagon inscribed in it 
A regular octagon and a regular dodecagon ($12$ sides) are both inscribed in a circle, with their vertices lining up whenever possible. If the largest distance between two points on the circumference is $16 \pi$, what is the radius of the circle?

The answer key says $96$ is the radius of the circle. That would mean the radius is bigger than the minor arc. Is that right?
There are 15 degree differences between the different sequence of points. $\dfrac{2\pi\cdot r\cdot15}{360}=16\pi$. unless you mean the large distance is the distance from $0$ to $360$ degrees, which implies radius is $8$.
 A: 
the largest distance between two points on the circumference

As I read this, it doesn't say the points are consecutive vertices of either polygon. (It technically doesn't say the points are any vertices of the polygons, either, but I suppose from the way the problem is worded that the "points" are the same ones we were trying to line up in the previous sentence.)
It also doesn't say how the distance is measured. Yes, it says "on the circumference," but that makes the most sense as a description of the location of the points.
If I meant to say that the measurement of distance should be the length of a circular arc lying on the circumference, I would use such explicit wording to say so, or I would say along the circumference.
By default, we measure distances between points as straight-line distance.
So we are looking at the farthest-apart vertices, which for the combined octagon-dodecagon will be two vertices at opposite ends of a diameter.
The distance between these points is the length of the diameter, $16\pi.$
The radius is half the diameter, $8\pi.$
Obviously this is an entirely different interpretation than was desired, but I blame poor wording of the question.

According to what I guess the author of the problem wanted to ask
(as opposed to what they actually did ask),
$16\pi$ is supposed to be the larger of the two possible arc lengths between consecutive vertices of either polygon along the circle.
The arc lengths are $2\pi r/12$ between consecutive vertices of the dodecagon when no vertex of the octagon lies between these vertices of the dodecagon
(in this case, one of the two dodecagon vertices coincides with a vertex of the octagon),
and $2\pi r/24$ between a vertex of the dodecagon and the vertex of the octagon when a vertex of the octagon lies between two consecutive vertices of the dodecagon.
The larger distance is $2\pi r/12,$ hence
\begin{align}
 2\pi r/12 &= 16\pi \\
 r/12 &= 8 && \text{Divide both sides by $2\pi.$} \\
 r &= 96  &&\text{Multiply both sides by $12.$} \\
\end{align}

Note that $2\pi/12 = \pi/6$ radians comes out to $30^\circ.$
Your $15^\circ$ angle is the smaller central angle between consecutive vertices along the circle.
And yes, either of these angles is less than one radian, hence the radius will be bigger than the arc length.
A: I believe the largest distance means the largest distance between two consecutive points is $16\pi$. Drawing a picture one can easily see that this is just the distance between two consecutive vertices of the dodecahedron. Therefore, we just multiply by $12$ to get the circumference and divide by $2\pi$ to get the radius, $96$.
