A mustache is created by taking the set of points $(x, y)$ in the $xy$-coordinate plane that satisfy $4 + 4 \cos(\pi x/24) \le y \le 6 + 6\cos(\pi x/24)$ and $-24 \le x \le 24$.  What is the area of the mustache?
To find the area of the mustache, we need to calculate the area between the curves \( y = 4 + 4 \cos\left(\frac{\pi x}{24}\right) \) and \( y = 6 + 6 \cos\left(\frac{\pi x}{24}\right) \) over the interval \( -24 \le x \le 24 \).

1. **Determine the difference between the upper and lower curves:**
   \[
   \Delta y = (6 + 6 \cos\left(\frac{\pi x}{24}\right)) - (4 + 4 \cos\left(\frac{\pi x}{24}\right))
   \]
   Simplifying this, we get:
   \[
   \Delta y = 6 + 6 \cos\left(\frac{\pi x}{24}\right) - 4 - 4 \cos\left(\frac{\pi x}{24}\right) = 2 + 2 \cos\left(\frac{\pi x}{24}\right)
   \]

2. **Set up the integral to find the area:**
   The area \( A \) is given by integrating the difference \( \Delta y \) over the interval \( -24 \le x \le 24 \):
   \[
   A = \int_{-24}^{24} (2 + 2 \cos\left(\frac{\pi x}{24}\right)) \, dx
   \]

3. **Split the integral into two parts:**
   \[
   A = \int_{-24}^{24} 2 \, dx + \int_{-24}^{24} 2 \cos\left(\frac{\pi x}{24}\right) \, dx
   \]

4. **Evaluate the first integral:**
   \[
   \int_{-24}^{24} 2 \, dx = 2 \left[ x \right]_{-24}^{24} = 2 (24 - (-24)) = 2 \cdot 48 = 96
   \]

5. **Evaluate the second integral:**
   \[
   \int_{-24}^{24} 2 \cos\left(\frac{\pi x}{24}\right) \, dx
   \]
   Since \( \cos\left(\frac{\pi x}{24}\right) \) is an even function, the integral over the symmetric interval \([-24, 24]\) will be zero:
   \[
   \int_{-24}^{24} \cos\left(\frac{\pi x}{24}\right) \, dx = 0
   \]
   Therefore,
   \[
   \int_{-24}^{24} 2 \cos\left(\frac{\pi x}{24}\right) \, dx = 2 \cdot 0 = 0
   \]

6. **Combine the results:**
   \[
   A = 96 + 0 = 96
   \]

The final answer is \(\boxed{96}\).