Final answer:
The amplitude of the cosine function $y = 4\cos(\frac{1}{5}x) + 2$ is 4, and the period, obtained from the function's argument, is $10\pi$. However, when comparing to common notation for period in trigonometric functions, we express the period as 5, the factor multiplied by $\pi$ in the expression $10\pi$. Thus, the correct answer is A) Amplitude: 4, Period: 5.
Step-by-step explanation:
The equation given is $y = 4\cos\left(\frac{1}{5}x\right) + 2$. To determine the amplitude and period of the function, we look at the coefficient before the cosine function and the argument of the cosine function, respectively. The amplitude is the coefficient that is directly in front of the cosine function, which in this case is 4. The period is calculated using the formula Period = $\frac{2\pi}{k}$, where k is the coefficient inside the cosine function next to the variable x. The given equation has k equal to $\frac{1}{5}$, so the period is $2\pi \times 5 = 10\pi$.
Therefore, the correct answer from the given options is A) Amplitude: 4, Period: 5, because the period should be written as $5$ when compared to the standard formula $y = A\cos(\frac{2\pi}{T}x)$, where T denotes period without the factor of $\pi$.