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 values of r, ω, and t. r = 8 in., ω = $\frac{pi}{3}$ radian per min, t = 12 min. (a) what is the angle generated by p in time t? θ = 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 = 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: Find the angle generated

Use the formula $\theta=\omega t$. Given $\omega = \frac{\pi}{3}$ rad/min and $t = 12$ min.
$\theta=\frac{\pi}{3}\times12 = 4\pi$ radians

Step2: Find the distance traveled

Use the arc - length formula $s = r\theta$. Since $r = 8$ in and $\theta=4\pi$ radians.
$s=8\times4\pi=32\pi$ inches

Step3: Find the linear speed

Use the formula $v=\frac{s}{t}$. We know $s = 32\pi$ inches and $t = 12$ min.
$v=\frac{32\pi}{12}=\frac{8\pi}{3}$ inches per minute

Answer:

(a) $4\pi$ radians
(b) $32\pi$ inches
(c) $\frac{8\pi}{3}$ inches per minute