91Ó°ÊÓ

To find the interval \([c, d]\) where the trajectory of \(\mathbf{R}\) is equivalent to the trajectory of \(\mathbf{r}\) over \([0, 2\pi]\), we set the position functions equal: $$\mathbf{r}(t) = \mathbf{R}(t)$$ So we have: $$\langle\cos t, 4 \sin t\rangle = (\cos 3t, 4 \sin 3t)$$ We can now solve for \(t\) when both position functions are equal components-wise: $$\cos t = \cos 3t$$ $$4 \sin t = 4 \sin 3t$$ For the first equality, we have $$t = \pi n, 3t = (2 m +1)\pi$$ For the second equality, we have $$\sin t = \sin 3t$$ It is clear that, the second equation must also hold for \(t = \pi n\), so we can rewrite \(3t = (2m+1)\pi\) as \(t = \frac{(2m +1)\pi}{3}\). Now, we have the interval \([0, 2\pi]\), then \(t = \frac{(2m +1)\pi}{3}\) must be a solution in that interval: $$0 \le \frac{(2m +1)\pi}{3} \le 2\pi$$ Solving for \(m\), $$m = 0, 1, 2$$ Thus, we have found three solutions for \(t\): $$t_0 = \frac{\pi}{3}, t_1 = \pi, t_2 = \frac{5\pi}{3}$$ For each \(t\) in the interval \([0, 2\pi]\), we can find the corresponding \(s\) in the interval \([c, d]\) as $$s = \frac{t}{3} \Rightarrow c = 0, d = \frac{2\pi}{3}$$

We will now find the velocity functions by calculating the derivatives of their respective position functions with respect to time. For \(\mathbf{r}(t)\): $$\mathbf{v}_r(t) = \frac{d}{dt}\langle\cos t, 4 \sin t\rangle = \langle-\sin t, 4\cos t\rangle$$ For \(\mathbf{R}(t)\): $$\mathbf{v}_R(t) = \frac{d}{dt}\langle\cos 3t, 4 \sin 3t\rangle = \langle-3\sin 3t, 12\cos 3t\rangle$$

The speed of an object is given by the magnitude of its velocity function. For \(\mathbf{r}(t)\): $$\text{speed}_r(t) = \vert\mathbf{v}_r(t)\vert = \sqrt{(-\sin t)^2 + (4\cos t)^2} = \sqrt{\sin^2 t + 16\cos^2 t}$$ For \(\mathbf{R}(t)\): $$\text{speed}_R(t) = \vert\mathbf{v}_R(t)\vert = \sqrt{(-3\sin 3t)^2 + (12\cos 3t)^2} = \sqrt{9\sin^2 3t + 144\cos^2 3t} = 3\sqrt{\sin^2 3t + 16\cos^2 3t}$$ Now graph the speeds of both objects with respect to time over their respective intervals: a) \(\text{speed}_r(t)\) over \([0, 2\pi]\) b) \(\text{speed}_R(t)\) over \([c, d]\) which is \([0, \frac{2\pi}{3}]\).