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.