Question

suppose that point p is on a circle with radius r, and ray op is rotating with angular speed ω. complete parts (a) through (c) for the given v. r = 4 in., ω = \\(\frac{\pi}{3}\\) radian per min, t = 6 min. (a) what is the angle generated by p in time t? θ = 2π radians (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.) (b) what is the distance traveled by p along the circle in time t? s = 8π inches (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.) (c) what is the linear speed of p? v = inches per minute (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Recall linear - speed formula

The formula for linear speed $v$ is $v=\frac{s}{t}$, where $s$ is the arc - length and $t$ is the time.

Step2: Identify known values

We know that $s = 8\pi$ inches (from part (b)) and $t = 6$ min.

Step3: Calculate linear speed

Substitute the values into the formula: $v=\frac{8\pi}{6}=\frac{4\pi}{3}$ inches per minute.

Answer:

$v=\frac{4\pi}{3}$ inches per minute