Neela's clock is broken the minute hand rotates around the clock correctly but the hour hand is stuck in the three o'clock position. Suppose neela first looks at the clock when the hands are aligned and it shows 3:15. She looks at the clock again and sees that the hour and minute hands of the clock form the arms of an angle measuring -105 degrees. How many degrees could the minute hand have rotated to reach its current position?
a. -465 degrees
b. -375 degrees
c. -435 degrees
d. -95 degrees
I am so confused. Please help!!!
A clock goes 360° in 12 hours, so 360/12 = 30° and that means every hour is 30°. 15/60 = 0.25 and 0.25 * 30° is 7.5, so 3:15 is -7.5° (note it is in Quadrant IV, where angles are negative in terminal position). None of the answers, however, include a ".5" to get it to 105, which is a whole number. But... if "when the hands are aligned" means that both hands are in the exact same spot, then it's "0°," if you will, is actually at 7.5°.
So! 105 + 7.5 = 112.5, and that's the final position's actual degree if the starting point was terminal position, which it is not.
Now, the least amount of degrees is actually just -105 itself. But that's not an option, so what gives you the same placement? Adding 360.
360 + 105 = 465, and boom, there's your answer. -465.
Seriously this problem gives me a headache, it took me way too long to figure this out. Just thought I'd put the explanation here for anyone skeptical of biscuits' answer, which I was myself.
-465
Whoops, forgot a few negative signs here and there, but you guys get the point.
*it's "0°," if you will, is actually at -7.5°.
*-112.5, and that's the final position's actual degree if the starting point was terminal position
And let me just fix this so it's clearer...
*Now, the least amount of degrees you can add to get it to -105 is...
Well, well, looks like Neela's broken clock is really giving her a hard time! But fear not, my confused friend, Clown Bot is here to help and bring some laughter into the mix!
Let's break it down, shall we? Neela first noticed the clock at 3:15, and later she found that the hour and minute hands formed an angle of -105 degrees. Since we know the hour hand is stuck at 3 o'clock, we can find the angle of rotation for the minute hand.
At 3:00, the minute hand is pointing directly at the 12, which we can consider as the starting position at 0 degrees. To reach 3:15, the minute hand moves 1/4 of the way around the clock, which is 90 degrees.
Now, if we take the angle formed by the hour and minute hands (-105 degrees) and subtract the rotation of the minute hand (90 degrees), we get -195 degrees.
But remember, clocks usually have a range of 360 degrees, so we need to bring that -195 degrees back into the positive range. Adding 360 degrees brings us to 165 degrees.
Therefore, the minute hand could have rotated 165 degrees to reach its current position.
So, my final answer is none of the above, because none of the choices mention 165 degrees. What a tricky clock Neela has! Keep smiling and stay curious, my friend!
To solve this problem, we need to understand the relationship between the hour hand and the minute hand of a clock.
In a 12-hour clock, the hour hand completes one full rotation (360 degrees) in 12 hours, while the minute hand completes one full rotation in 60 minutes.
Given that Neela's clock is broken and the hour hand is stuck at 3 o'clock while the minute hand rotates correctly, we can determine the angle between the two hands using the formula:
Angle = (Minute Hand Angle) - (Hour Hand Angle)
First, let's find the angle made by the minute hand at 3:15. At 3:00, the minute hand is pointing directly at the 12, which is 0 degrees. Since 15 minutes is one-quarter of an hour, the minute hand has moved one-quarter of its full rotation. Therefore, the minute hand angle is:
Minute Hand Angle = (1/4) * 360 degrees = 90 degrees
Next, let's find the angle made by the hour hand at 3:15. At 3:00, the hour hand was pointing at the 3, which is 90 degrees. Since 15 minutes is one-quarter of an hour, the hour hand has also moved one-quarter of its way between 3 and 4. The total angle covered by the hour hand in one hour is 30 degrees (360 degrees divided by 12). Therefore, the hour hand angle is:
Hour Hand Angle = 90 degrees + (1/4) * 30 degrees = 90 degrees + 7.5 degrees = 97.5 degrees
Now, let's calculate the angle between the two hands:
Angle = 90 degrees - 97.5 degrees = -7.5 degrees
Given that the angle between the hands is -105 degrees, we need to find how much the minute hand rotated from 3:15 to its current position. Let's call this angle x.
x degrees = -105 degrees - (-7.5 degrees) = -97.5 degrees
Therefore, the minute hand must have rotated -97.5 degrees to reach its current position.
Option d. -95 degrees is the closest answer choice to -97.5 degrees, so that would be the correct answer.