use a graphing utility to graph the polar equation. r = 3 sin(5θ/2) find an interval for θ for which the…

Explanation:

Step1: Recall polar - function periodicity

For a polar function of the form $r = A\sin(n\theta)$ or $r=A\cos(n\theta)$, the period $T$ is given by the formula. When $n$ is odd, the period of $r = A\sin(n\theta)$ is $T = \pi$. When $n$ is even, the period of $r = A\sin(n\theta)$ is $T = 2\pi$. In the given function $r = 3\sin\left(\frac{5\theta}{2}\right)$, here $n=\frac{5}{2}$. The general formula for the period of $y = a\sin(b\theta)$ is $T=\frac{2\pi}{|b|}$. For $r = 3\sin\left(\frac{5\theta}{2}\right)$, we have $b = \frac{5}{2}$.

Step2: Calculate the period

Using the formula $T=\frac{2\pi}{|b|}$, substituting $b=\frac{5}{2}$, we get $T=\frac{2\pi}{\frac{5}{2}}=\frac{4\pi}{5}$. But we want to find the interval for which the entire graph is traced once. For a polar - function $r = A\sin(n\theta)$ with non - integer $n$, we consider the fact that the function will repeat its values. The function $r = 3\sin\left(\frac{5\theta}{2}\right)$ will trace the entire graph when $\theta$ varies from $0$ to $4\pi$. This is because as $\theta$ goes from $0$ to $4\pi$, the argument of the sine function $\frac{5\theta}{2}$ goes from $0$ to $10\pi$.

Answer:

C. $[0,4\pi)$