the polar function r = f(θ), where f(θ) = 3 sin θ, is graphed in the polar coordinate system on the interval…

the polar function r = f(θ), where f(θ) = 3 sin θ, is graphed in the polar coordinate system on the interval 0 < θ < π/2. which of the following statements is true about the distance between the origin and the point on the graph with polar coordinates (f(θ),θ)? a the distance is increasing, because f(θ) is positive and increasing on this interval. b the distance is decreasing, because f(θ) is positive and decreasing on this interval.

the polar function r = f(θ), where f(θ) = 3 sin θ, is graphed in the polar coordinate system on the interval 0 < θ < π/2. which of the following statements is true about the distance between the origin and the point on the graph with polar coordinates (f(θ),θ)? a the distance is increasing, because f(θ) is positive and increasing on this interval. b the distance is decreasing, because f(θ) is positive and decreasing on this interval.

Answer

Explanation:

Step1: Recall polar - distance relationship

In polar coordinates, the distance from the origin to the point $(r,\theta)$ is given by $r = f(\theta)$. Here $f(\theta)=3\sin\theta$.

Step2: Analyze the derivative of $y = \sin\theta$

The derivative of $y = \sin\theta$ is $y'=\cos\theta$. For $\theta\in(0,\frac{\pi}{2})$, $\cos\theta> 0$.

Step3: Analyze the function $f(\theta)$

Since $f(\theta)=3\sin\theta$ and the derivative of $\sin\theta$ is $\cos\theta>0$ for $\theta\in(0,\frac{\pi}{2})$, and the coefficient $3>0$, the function $f(\theta)$ is positive and increasing on the interval $(0,\frac{\pi}{2})$.

Answer:

A. The distance is increasing, because $f(\theta)$ is positive and increasing on this interval.