Question

suppose the tip of the minute hand of a clock is 12 in. from the center of the clock. for 40 min the duration, determine the distance traveled by the tip of the minute hand. the distance traveled by the tip of the minute hand is (simplify your answer. type your answer in terms of π. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Find the angle the minute - hand rotates

The minute - hand of a clock makes a full rotation ($360^{\circ}$ or $2\pi$ radians) in 60 minutes. For 40 minutes, the angle $\theta$ it rotates is calculated as a proportion.
$\theta=\frac{40}{60}\times2\pi=\frac{4\pi}{3}$ radians.

Step2: Use the arc - length formula

The arc - length formula is $s = r\theta$, where $r$ is the radius of the circle and $\theta$ is the central angle in radians. Given that $r = 12$ inches and $\theta=\frac{4\pi}{3}$ radians.
$s=12\times\frac{4\pi}{3}$

Step3: Simplify the expression

$12\times\frac{4\pi}{3}=\frac{48\pi}{3}=16\pi$ inches.

Answer:

$16\pi$ inches