Given that $x = 5\sin(3t), t\ge 0$: Find the maximum value of $x$ and the smallest value of $t$ for which it occurs.

Question

Given that: $$x = 5\sin(3t), t \ge 0$$

Find the maximum value of $x$ and the smallest value of $t$ for which it occurs.

I have figured out the smallest value by:

$$\frac{dx}{dt}=15\cos(3t)$$

when $$15\cos(3t) = 0$$

gives us the smallest values of $t$: 30 degrees or $\frac{\pi}{6}$

From the provided answers, the maximum value of $x$ is 5, but I'm not sure how to obtain this. Any hints would be appreciated!

Answer 1

The maximum value of $\sin 3t$ is $1$. So the maximum value of $5\sin 3t$ is $5$.

Answer 2

$\sin x\leq 1$ and the smallest positive value of $x$ where $\sin x $ becomes $1$ is $\frac {\pi} 2$. Just put $x=3t$ in this and multiply by $5$.