Let's draw a picture of the problem:
From the given picrture, we can make the following right triangle:
where r denotes the radius of the small circle. Then, by applying Pythagorean theorem, we have

which gives

then, the radius of the small circle is given by
![\begin{gathered} r=\sqrt[]{75} \\ r=8.66\text{ in} \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/x56pes96tn2aqbqg3w9ig4bkkmdp1ywvqi.png)
Therefore, the first answer is r=8.66 inches.
Now, let's find the great circle distance from the pole to its circunference. So, let's make a picture of the problem:
where d is the distance from the pole to the greate circle. So, we have another right triangle:
So, by applying Pythagoren theorem, we have

which gives
![\begin{gathered} d^2=25+100 \\ d^2=125 \\ \text{then} \\ d=\sqrt[]{125} \\ d=11.18\text{ in} \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/tz81xt58865vcrxs7809y7wfeujy5k6nlm.png)
Therefore, the answer for the second question is d = 11.18 inches