Question

the minute hand of a clock is 8 inches long and moves from 12 to 8 oclock. how far does the tip of the minute hand move? express your answer in terms of π and then round to two decimal places.
the minute hand moves inches.
(simplify your answer. type your answer in terms of π. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Determine the fraction of the circle

A clock is a circle (360 degrees), and there are 12 hours. From 12 to 8, the minute hand moves 8 out of 12 hours. So the fraction is $\frac{8}{12}=\frac{2}{3}$.

Step2: Recall the arc length formula

The arc length formula is $s = r\theta$, where $r$ is the radius and $\theta$ is the central angle in radians. For a full circle, $\theta = 2\pi$, so the arc length of a full circle (circumference) is $C = 2\pi r$. Using the fraction of the circle, the arc length $s$ is the fraction times the circumference.

Step3: Calculate the arc length

The radius $r$ of the minute hand's path is 8 inches. The circumference is $2\pi r = 2\pi\times8 = 16\pi$. The fraction of the circle is $\frac{2}{3}$, so the arc length $s=\frac{2}{3}\times16\pi=\frac{32}{3}\pi$.

Answer:

$\frac{32}{3}\pi$