Maximize velocity with parametric equations

Question

Suppose we are asked to find the value of t at which an object is at its maximum velocity, if it travels on a path governed by:

$x = 2 + 8\cos(t)$

$y = \sin(t)$

Here's what I understand:

$\frac{dx}{dt} = -8\sin(t)$

$\frac{dy}{dt} = \cos(t)$

The velocity at a given value of t would be:

$\|v\| = \sqrt{(\frac{dy}{dt})^2 + (\frac{dx}{dt})^2}$

And if we wanted to find the value of t with the maximum velocity, we could take the derivative, set it to zero, and solve. we can ignore the square root because it is maximized when its inside is maximized:

$\|v\| = 64\sin^2(t) + \cos^2(t)$

$\|v\|' = 128\sin(t)\cos(t) - 2\sin(t)\cos(t)$

$0 = 126\sin(t)\cos(t)$

$t = a\sin(0)$ OR $a\cos(0)$ = $\frac{\pi}{2}n$

But that doesn't make sense. Did I do something wrong, or is that a logical answer?

Answer 1

Your answer is kind of correct, but here's a faster way of getting it: $$|v|^2=64\sin^2t+\cos^2t=64-63\cos^2t\implies \color{red}{|v|\le 8}$$ The equality holds when $\cos t=0\implies t=(2n-1)\frac{\pi}2$