Question

through how many radians does the minute hand of a clock rotate from 12:40 pm to 1:15 pm?

the minute hand rotates through \\(\square\\) radians.
(simplify your answer. type an exact answer in terms of \\(\pi\\). use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Calculate time difference

From 12:40 PM to 1:15 PM, the time elapsed is \(75 - 40 = 35\) minutes? Wait, no: 1:15 is 75 minutes past 12, 12:40 is 40 minutes past 12, so \(75 - 40 = 35\)? Wait, no, 1:15 is 60 + 15 = 75 minutes, 12:40 is 40 minutes, so difference is \(75 - 40 = 35\) minutes? Wait, no, wait: 12:40 to 1:00 is 20 minutes, 1:00 to 1:15 is 15 minutes, so total is \(20 + 15 = 35\) minutes? Wait, no, wait, 12:40 to 1:15: 1:15 - 12:40 = 35 minutes. Wait, but the minute hand makes a full rotation (360 degrees or \(2\pi\) radians) in 60 minutes. So the rate is \(\frac{2\pi}{60}\) radians per minute, which simplifies to \(\frac{\pi}{30}\) radians per minute.

Step2: Calculate radians rotated

Time elapsed is 35 minutes? Wait, no, wait: 12:40 to 1:15: let's check again. 12:40 to 1:15: 1:15 is 75 minutes after 12, 12:40 is 40 minutes after 12, so 75 - 40 = 35 minutes? Wait, no, 12:40 to 1:15: 1:15 - 12:40 = 35 minutes. Wait, but let's confirm: 60 - 40 = 20 (from 12:40 to 1:00), then 15 minutes from 1:00 to 1:15, so 20 + 15 = 35. So time elapsed \(t = 35\) minutes. The rate is \(\frac{2\pi}{60}\) radians per minute (since 60 minutes for a full circle). So radians rotated \(= \text{rate} \times \text{time} = \frac{2\pi}{60} \times 35\). Simplify: \(\frac{2\pi \times 35}{60} = \frac{70\pi}{60} = \frac{7\pi}{6}\). Wait, wait, that can't be. Wait, no, wait, maybe I made a mistake in time elapsed. Wait, 12:40 to 1:15: let's count the minutes. 12:40, 12:41,..., 1:15. Let's do 1:15 is 75 minutes, 12:40 is 40 minutes, so 75 - 40 = 35 minutes. Wait, but let's check the rate again: full rotation (2π radians) in 60 minutes, so per minute, \(\frac{2\pi}{60} = \frac{\pi}{30}\) radians per minute. So 35 minutes: \(35 \times \frac{\pi}{30} = \frac{35\pi}{30} = \frac{7\pi}{6}\). Wait, but let's check again. Wait, maybe I messed up the time. Wait, 12:40 to 1:15: 1:15 is 1 hour and 15 minutes, which is 75 minutes, 12:40 is 40 minutes, so 75 - 40 = 35 minutes. So yes, 35 minutes. Then \(\frac{\pi}{30}\) radians per minute times 35 minutes: \(\frac{35\pi}{30} = \frac{7\pi}{6}\). Wait, but let's confirm with another approach. The minute hand moves 6 degrees per minute (since 360/60 = 6). So 35 minutes: 35 6 = 210 degrees. Convert 210 degrees to radians: 210 (\(\pi\)/180) = (7π)/6. Yes, that's correct. So the minute hand rotates through \(\frac{7\pi}{6}\) radians? Wait, no, wait, wait: 210 degrees is \(\frac{7\pi}{6}\) radians? Wait, 180 degrees is π, 210 - 180 = 30, so 210 degrees is π + π/6 = 7π/6. Yes. So that's correct.

Wait, but wait, maybe I made a mistake in time elapsed. Let's check again: 12:40 to 1:15. 12:40 to 1:00 is 20 minutes, 1:00 to 1:15 is 15 minutes, so 20 + 15 = 35 minutes. Correct. So time is 35 minutes. Rate is 2π radians per 60 minutes, so per minute, π/30 radians. So 35 * π/30 = 35π/30 = 7π/6. Yes.

Answer:

\(\boxed{\dfrac{7\pi}{6}}\)