Question

what are the amplitude and period of $y = 4\cos\left(\frac{x}{5}\
ight)$?\
\
the amplitude is 8 and the period is $\frac{2\pi}{5}$.\
the amplitude is 8 and the period is $10\pi$.\
the amplitude is 4 and the period is $\frac{2\pi}{5}$.\
the amplitude is 4 and the period is $10\pi$.

Explanation:

Step1: Recall the formula for amplitude and period

For a cosine function \(y = A\cos(Bx)\), the amplitude is \(|A|\) and the period is \(T=\frac{2\pi}{|B|}\).

Step2: Identify \(A\) and \(B\) in the given function

In the function \(y = 4\cos(\frac{x}{5})\), we have \(A = 4\) and \(B=\frac{1}{5}\).

Step3: Calculate the amplitude

The amplitude \(|A|=|4| = 4\).

Step4: Calculate the period

Using the formula \(T=\frac{2\pi}{|B|}\), substitute \(B=\frac{1}{5}\). Then \(T=\frac{2\pi}{\frac{1}{5}}=10\pi\).

Answer:

The amplitude is \(4\) and the period is \(10\pi\) (i.e., the first option).