https://file.aiquickdraw.com/mathbot/file/1729794441439dr6l66lu.jpg

The given function is:

\[
y = 5 \cos\left(\frac{x}{4}\right)
\]

Let's break down the key components of this equation to find the **amplitude** and **period**.

### 1. **Amplitude**:
In a cosine function of the form \( y = A \cos(Bx) \), the amplitude is the absolute value of \( A \). 

Here, \( A = 5 \), so the **amplitude** is \( 5 \).

### 2. **Period**:
The period of a cosine function is given by the formula:

\[
\text{Period} = \frac{2\pi}{|B|}
\]

In the function \( y = 5 \cos\left(\frac{x}{4}\right) \), \( B = \frac{1}{4} \), so the period is:

\[
\text{Period} = \frac{2\pi}{\frac{1}{4}} = 8\pi
\]

### Conclusion:
- **Amplitude**: 5
- **Period**: \( 8\pi \)

Thus, the correct answer is: **The amplitude is 5 and the period is \( 8\pi \)**, which corresponds to the third option in the choices.

Let me know if you need further details or explanations. Here are some related questions to expand on this topic:

1. How do you determine the phase shift in a cosine function?
2. What is the difference between amplitude and vertical shift in trigonometric functions?
3. Can the amplitude of a trigonometric function ever be negative?
4. How does changing the value of \( B \) affect the period of a cosine function?
5. How would the graph of this function look, and how do we interpret its key features?

**Tip**: Always remember that the amplitude only affects the vertical stretch, while the period is determined by how fast the function oscillates horizontally.

What are the amplitude and period of y = 5 cos(x/4)?

Find the amplitude and period of the trigonometric function y = 5 cos(x/4). The amplitude is 5, and the period is 8π. This problem involves cosine function properties and is suitable for high school students in Grades 9-12.

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