Question

on an analog clock, what is the measure of the minor arc between 6:00 and 9:00 in radians? you may use the clock below as a reference.

Explanation:

Step1: Find the angle per hour mark

A clock is a circle, so it has \(2\pi\) radians. There are 12 hours, so the angle between each hour mark is \(\frac{2\pi}{12}=\frac{\pi}{6}\) radians per hour.

Step2: Find the number of hours between 6 and 9

The number of hours between 6 and 9 is \(9 - 6 = 3\) hours.

Step3: Calculate the arc length in radians

Multiply the number of hours by the angle per hour: \(3\times\frac{\pi}{6}=\frac{\pi}{2}\) radians. Wait, no, wait. Wait, the minor arc between 6 and 9: wait, 6 to 9 is 3 hours, but wait, a clock is 12 hours, so the angle between 6 and 9: each hour is 30 degrees (since 360/12=30), so 3*30=90 degrees, which is \(\frac{\pi}{2}\) radians? Wait, no, wait, 6 to 9: 6 is at the bottom, 9 is at the left. The angle between them: from 6 to 9, that's 3 hours. Wait, but let's re - calculate. The formula for the central angle between two hours \(h_1\) and \(h_2\) is \(\theta=\frac{2\pi}{12}\times|h_1 - h_2|\) (for the minor arc). So \(|9 - 6| = 3\), so \(\theta=\frac{2\pi}{12}\times3=\frac{\pi}{2}\)? Wait, no, \(\frac{2\pi}{12}=\frac{\pi}{6}\) per hour, times 3 hours is \(\frac{3\pi}{6}=\frac{\pi}{2}\). But wait, 6 to 9: 6 is 180 degrees from 12, 9 is 270 degrees from 12. The difference is 90 degrees, which is \(\frac{\pi}{2}\) radians. But wait, maybe I made a mistake. Wait, no, 6 o'clock: the hour hand is at 6, which is \(\pi\) radians from 12 (if 12 is 0 radians). 9 o'clock: the hour hand is at 9, which is \(\frac{3\pi}{2}\) radians from 12. The difference is \(\frac{3\pi}{2}-\pi=\frac{\pi}{2}\) radians. So the minor arc between 6 and 9 is \(\frac{\pi}{2}\) radians? Wait, no, wait, 6 to 9: 6 is at \(\pi\) (if 12 is 0), 9 is at \(\frac{3\pi}{2}\). The distance between them is \(\frac{3\pi}{2}-\pi=\frac{\pi}{2}\), but also, the other arc (major arc) would be \(2\pi-\frac{\pi}{2}=\frac{3\pi}{2}\), so the minor arc is \(\frac{\pi}{2}\). Wait, but let's check again. Wait, 12 hours correspond to \(2\pi\) radians. So each hour mark is \(\frac{2\pi}{12}=\frac{\pi}{6}\) radians apart. The number of hour marks between 6 and 9 is 3 (6 to 7, 7 to 8, 8 to 9). So the central angle is \(3\times\frac{\pi}{6}=\frac{\pi}{2}\) radians. Yes, that's correct.

Answer:

\(\boxed{\dfrac{\pi}{2}}\)