#1. An accurate clock shows 8 o’clock in the morning. Through how may degrees will the hour hand rotate when the clock shows 2 o’clock in the afternoon? Step 1: Determine the total time difference Step 2: Calculate how many degrees the hour hand moves per hour Step 3: Calculate the total degrees of rotation for 6 hours
A. 144º
B. 150º
C. 168º
D. 180º
Solution:
To calculate how many degrees the hour hand will rotate between 8:00 AM and 2:00 PM, we can follow these steps:
From 8:00 AM to 2:00 PM is a difference of 6 hours.
The clock is divided into 12 hours, and the hour hand moves 360 degrees in a full rotation (i.e., over 12 hours). Therefore, the hour hand moves:
\[
\frac{360 \text{ degrees}}{12 \text{ hours}} = 30 \text{ degrees per hour}.
\]
Since the hour hand moves 30 degrees per hour, in 6 hours, the total rotation will be:
\[
6 \times 30 = 180 \text{ degrees}.
\] Final Answer:
The hour hand will rotate 180 degrees from 8:00 AM to 2:00 PM.
#2. The reflex angle between the hands of a clock at 10.25 is:
A. 180º
B.192\( \frac{ 1º}{2} \)
C. 195º
D.197.5º
Solution:
To calculate the reflex angle between the hands of a clock at 10:25, we need to follow these steps:
Step 1: Calculate the angle of the hour hand
– The hour hand moves 30 degrees per hour (since \( \frac{360^\circ}{12} = 30^\circ \)).
– At 10:00, the hour hand is at \( 10 \times 30 = 300^\circ \) from the 12:00 position.
– In 25 minutes, the hour hand will move slightly further. Since the hour hand moves 30 degrees every hour (60 minutes), in 25 minutes, it will move:
\[
\frac{25}{60} \times 30 = 12.5^\circ.
\] – So, at 10:25, the hour hand will be at:
\[
300^\circ + 12.5^\circ = 312.5^\circ.
\]
Step 2: Calculate the angle of the minute hand
– The minute hand moves 6 degrees per minute (since \( \frac{360^\circ}{60} = 6^\circ \)).
– At 25 minutes, the minute hand will be at:
\[
25 \times 6 = 150^\circ.
\]
Step 3: Calculate the smaller angle between the hour and minute hands
The smaller angle between the hour and minute hands is the absolute difference between their positions:
\[
\left| 312.5^\circ – 150^\circ \right| = 162.5^\circ.
\]
Step 4: Calculate the reflex angle
The reflex angle is the larger angle, which is the remaining angle in the circle:
\[
360^\circ – 162.5^\circ = 197.5^\circ.
\]
Final Answer:
The reflex angle between the hands of the clock at 10:25 is 197.5 degrees.