What is the degree measure of the acute angle formed by the hour and minute hands at 2:16?
let's consider "noon" on the clock as 0°, then 16 minutes will be at 96°
2:16 = 2 16/60 of an hour = 34/15 hours
each hour is represented by 30°
so the hour hand has moved (34/15)(30) or 68°
which means that the acute angle between the hands
= 96-68 °
= 28°
To determine the degree measure of the acute angle formed by the hour and minute hands at 2:16, we need to calculate the positions of both hands.
1. Hour hand: At 2:16, the hour hand will have moved a fraction of the distance between the 2 and 3 o'clock positions. Since there are 12 hours on a clock, each hour mark is 30 degrees apart (360 degrees divided by 12). The minute hand also moves, causing the hour hand to move slightly more than one-sixth of an hour. The angle moved by the hour hand is given by (1/6) * 30 degrees.
2. Minute hand: At 2:16, the minute hand will be at the 16-minute mark. There are 60 minutes on a clock, and since the entire circumference of a clock is 360 degrees, each minute mark is 6 degrees apart (360 degrees divided by 60). So, the angle rotated by the minute hand is 16 * 6 degrees.
Now, let's calculate the positions of both hands.
1. Hour hand: (1/6) * 30 degrees = 5 degrees
2. Minute hand: 16 minutes * 6 degrees = 96 degrees
To find the acute angle between the hour and minute hands, we take the absolute difference between the two angles and subtract it from 360 degrees (since the hands are moving in a clockwise direction).
Acute angle = 360 degrees - |5 degrees - 96 degrees|
Acute angle = 360 degrees - |-91 degrees|
Acute angle = 360 degrees - 91 degrees
Acute angle = 269 degrees
Therefore, at 2:16, the acute angle formed by the hour and minute hands is 269 degrees.
To calculate the degree measure of the acute angle formed by the hour and minute hands at 2:16, we need to determine the positions of the hour and minute hands.
Step 1: Convert the time to positions on the clock:
The hour hand moves 30 degrees per hour and 0.5 degrees per minute. At 2:16, the hour hand would be pointing between the 2 and 3 hours mark. To calculate the exact position, we know that from 2:00 to 3:00, the hour hand moves 30 degrees. Since 16 minutes have passed, the additional movement can be calculated as (16 minutes * 0.5 degrees per minute = 8 degrees). Therefore, the hour hand is at (30 degrees + 8 degrees = 38 degrees).
The minute hand moves 6 degrees per minute. At 16 minutes past the hour, the minute hand would be at (16 minutes * 6 degrees per minute = 96 degrees).
Step 2: Calculate the angle between the hands:
To find the acute angle formed by the hands, we need to determine the difference in their positions. In this case, the hour hand is at 38 degrees, and the minute hand is at 96 degrees. The absolute difference between these two positions is (96 degrees - 38 degrees = 58 degrees).
Step 3: Adjust for smaller angle:
Since the angle between the hour and minute hands could be greater than 180 degrees, we need to check if the angle obtained is larger than 180 degrees. If it is, we subtract it from 360 degrees to get the smaller angle. In this case, the angle is smaller than 180 degrees, so we don't need to make any adjustments.
Therefore, the degree measure of the acute angle formed by the hour and minute hands at 2:16 is 58 degrees.