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Let $\mathcal{G}$ be the graph of the parametric equations \begin{align*} x &= \cos(4t),\\ y &= \sin(6t). \end{align*}What is the length of the smallest interval $I$ such that the graph of these equations for all $t\in I$ produces the entire graph $\mathcal{G}$?

I thought it was a pretty straightforward question, since $x = \cos(4t)$ had a period of $\frac{\pi}{2}$ and $y = \sin(6t)$ had a period of $\frac{\pi}{3}$, the LCM would be $\boxed{\pi}$, hence the interval.

However, the answer keeps on coming as incorrect. Am I overlooking something here?

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    The term *graph* is kind of inappropriate here, since the set $\{(\cos(4t),\sin(6t)),\;t\in[0,\pi]\}$ is not (globally) a graph. Additionally, what should it be the correct answer, if not $\pi$? – 2017-02-01
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    Unfortunately, the system this question is on does not tell you the correct answer. That was why I asked it here, because I wasn't sure what the answer would be... @JackD'Aurizio – 2017-02-01
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    Apart from the debatable wording of the question, I agree that $\pi$ is the wanted minimal length. – 2017-02-01

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Evidently by graph they mean the set of points in the coordinate plane which is a curved path from $(-1,1)$ to $(-1,-1)$ in which case the answer would be $\dfrac{\pi}{2}$ since between $t=-\frac{\pi}{4}$ and $t=\frac{\pi}{4}$ the entire set of $(x,y)$ coordinates in the path has been "covered" although this is only half the path.

  1. For $t\in \left[-\frac{\pi}{4},-\frac{\pi}{6}\right]$ the path goes from $(-1,1)$ to the point $\left(-\frac{1}{2},0\right)$.
  2. For $t\in \left[-\frac{\pi}{6},0\right]$ the path goes from $\left(-\frac{1}{2},0\right)$ to the point $(1,0)$.
  3. For $t\in \left[0,\frac{\pi}{6}\right]$ the path goes from $(1,0)$ to the point $\left(-\frac{1}{2},0\right)$.
  4. For $t\in \left[\frac{\pi}{6},\frac{\pi}{4}\right]$ the path goes from $\left(-\frac{1}{2},0\right)$ to the point $(-1,-1)$.

parametric graph

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    Ah, so the graph actually had a literal meaning here. Thanks, that helped! – 2017-02-01
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    Yes, this seems to be a classic "trick question." Rather than simply asking for the period of the path, which is clearly $\pi$, they speak of the length of $t$ interval and "graph." And by "graph" evidently they mean the projection of the path onto the Cartesian plane. Since it traces back over the same "graph" twice during one period the number that is sought is evidently half the period. – 2017-02-01