https://file.aiquickdraw.com/mathbot/file/1725913787311hqda3z20.jpg

To solve this problem, we need to find the region enclosed between the two curves and sketch it.

The two curves are given as:
1. \( y = 5 \cos(5x) \)
2. \( y = 5 - 5 \cos(5x) \)

### Step-by-step approach:

1. **Determine the points of intersection:**
   To find the enclosed region, we first need to determine where the two curves intersect. To do this, we set the two equations equal to each other:

   \[
   5 \cos(5x) = 5 - 5 \cos(5x)
   \]

   Simplifying this:

   \[
   2 \cdot 5 \cos(5x) = 5
   \]

   \[
   \cos(5x) = \frac{1}{2}
   \]

   The general solution for \( \cos(5x) = \frac{1}{2} \) is:

   \[
   5x = \pm \frac{\pi}{3} + 2n\pi \quad \text{(where \(n\) is any integer)}
   \]

   Solving for \(x\):

   \[
   x = \pm \frac{\pi}{15} + \frac{2n\pi}{5}
   \]

   Considering the range \( 0 \leq x \leq \frac{\pi}{5} \), the solutions for \(x\) are:

   \[
   x = \frac{\pi}{15}
   \]

2. **Sketch the region:**
   We now know that the curves intersect at \( x = \frac{\pi}{15} \). To sketch the region between the curves, plot the curves \( y = 5 \cos(5x) \) and \( y = 5 - 5 \cos(5x) \) from \( x = 0 \) to \( x = \frac{\pi}{5} \), and shade the area between them.

### Would you like me to generate the sketch for you or continue with further details?

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### Related Questions:
1. How do you calculate the area enclosed between two curves?
2. What is the period of the function \( y = 5 \cos(5x) \)?
3. How do you handle a trigonometric equation like \( \cos(5x) = \frac{1}{2} \)?
4. What is the significance of points of intersection when calculating the enclosed area?
5. How would you generalize the process of finding the enclosed area for more complex functions?

#### Tip:
Always check the points of intersection before proceeding with further analysis such as area calculations.

Sketch the region enclosed by the curves: y = 5cos(5x) and y = 5 - 5cos(5x), for 0 ≤ x ≤ π/5.

Learn how to find the points of intersection and sketch the region enclosed by the curves y = 5cos(5x) and y = 5 - 5cos(5x) in the interval 0 ≤ x ≤ π/5. This problem involves trigonometry, cosine functions, and curve sketching. Suitable for students in Grade 11-12.

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