The diagram below shows a shaded region bounded by two concentric circles where the outer circle has twice the radius of the inner circle. The total boundary of the shaded region has length $36\pi$. Find $n$ such that the area of the shaded region is $n\pi$.
[img]https://cdn.artofproblemsolving.com/attachments/4/5/c9ffdc41c633cc61127ef585a45ee5e6c0f88d.png[/img]
1. Let the radius of the smaller circle be \( r \). The radius of the larger circle is then \( 2r \) since the outer circle has twice the radius of the inner circle.

2. The total boundary length of the shaded region is given as \( 36\pi \). This boundary consists of the circumference of the inner circle and the circumference of the outer circle. Therefore, we can write:
   \[
   2\pi r + 2\pi (2r) = 36\pi
   \]

3. Simplify the equation:
   \[
   2\pi r + 4\pi r = 36\pi
   \]
   \[
   6\pi r = 36\pi
   \]

4. Divide both sides by \( 6\pi \):
   \[
   r = 6
   \]

5. Now that we have found the radius of the smaller circle \( r = 6 \), we can find the area of the shaded region. The area of the shaded region is the difference between the area of the larger circle and the area of the smaller circle.

6. The area of the larger circle (radius \( 2r = 12 \)) is:
   \[
   \pi (2r)^2 = \pi (12)^2 = 144\pi
   \]

7. The area of the smaller circle (radius \( r = 6 \)) is:
   \[
   \pi r^2 = \pi (6)^2 = 36\pi
   \]

8. The area of the shaded region is:
   \[
   144\pi - 36\pi = 108\pi
   \]

9. Therefore, \( n = 108 \).

The final answer is \( \boxed{ n = 108 } \).