1. **State the problem:** Find the period of the function $y = 4 \cos(5x)$.\n\n2. **Recall the formula for the period of cosine:** The general form is $y = A \cos(Bx)$, where the period $T$ is given by $$T = \frac{2\pi}{|B|}.$$\n\n3. **Identify $B$ in the given function:** Here, $B = 5$.\n\n4. **Calculate the period:** Substitute $B = 5$ into the formula:\n$$T = \frac{2\pi}{5}.$$\n\n5. **Interpretation:** The period of $y = 4 \cos(5x)$ is $\frac{2\pi}{5}$. This means the function completes one full cycle every $\frac{2\pi}{5}$ units along the x-axis.\n\n**Final answer:** $\boxed{\frac{2\pi}{5}}$