consider the polar curve $r = 4\\sin(2\\theta)$. what is the length of each petal, and how many petals does…

Explanation:

Step1: Recall polar - rose curve properties

For a polar curve of the form $r = a\sin(n\theta)$ or $r=a\cos(n\theta)$. If $n$ is even, the number of petals is $2n$; if $n$ is odd, the number of petals is $n$. Here $r = 4\sin(2\theta)$, and $n = 2$ (even), so the number of petals is $2n=4$.

Step2: Find the length of each petal

The maximum value of $r$ gives the length of each petal. For $y = A\sin(Bx)$, the maximum value is $|A|$. In the polar - curve $r = 4\sin(2\theta)$, the maximum value of $r$ occurs when $\sin(2\theta)=1$. So, $r_{max}=4$.

Answer:

The length of each petal is 4, and the number of petals is 4.