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 $\mathcal{G}$?

The answer is not pi. I have tried that, and it says it is wrong.

\pi/2 trust me 100% correct

It’s pi/2 trust him

To find the length of the smallest interval $I$ that generates the entire graph of $\mathcal{G}$, we need to consider the range of values for $t$ that produce the complete graph.

The graph of $\mathcal{G}$ is a closed curve formed by the parametric equations $x = \cos(4t)$ and $y = \sin(6t)$. This means that the values of $x$ and $y$ repeat periodically as $t$ varies.

Let's analyze the periods of both $\cos(4t)$ and $\sin(6t)$ separately:

For $\cos(4t)$, the period is obtained by dividing the period of the cosine function, which is $2\pi$, by the coefficient in front of $t$. In this case, the coefficient is 4, so the period of $\cos(4t)$ is $\frac{2\pi}{4} = \frac{\pi}{2}$.

For $\sin(6t)$, the period is calculated in a similar way. The period of the sine function is $2\pi$, and the coefficient in front of $t$ is 6. Thus, the period of $\sin(6t)$ is $\frac{2\pi}{6} = \frac{\pi}{3}$.

To generate the entire graph of $\mathcal{G}$, we need $t$ to cover a range that includes both of these periods. The smallest interval that satisfies this condition is the least common multiple (LCM) of the two periods.

To find the LCM of $\frac{\pi}{2}$ and $\frac{\pi}{3}$, we can convert these fractions to a common denominator of 6.

$\frac{\pi}{2} = \frac{3\pi}{6}$

$\frac{\pi}{3} = \frac{2\pi}{6}$

Now, the LCM of $3\pi$ and $2\pi$ is $6\pi$.

Thus, the smallest interval $I$ that produces the entire graph of $\mathcal{G}$ is $I = [t_1, t_2]$ such that $t_1 = 0$ and $t_2 = 6\pi$. So, the length of the interval $I$ is $t_2 - t_1 = 6\pi - 0 = \boxed{6\pi}$.

Note: It is important to remember that when working with periodic functions like sine and cosine, the periods determine the range of values for $t$ that result in the complete graph. In this case, the graph of $\mathcal{G}$ is a closed curve, so we need to consider the least common multiple of the periods of the individual functions.

Hmmm. You're right. It's 2π/3