PLEASE HELP WITH MATH ANY HELP OR ANSWER IS APPRECIATED!
1. If the hands on a round clock moved counter-clockwise, to what number would the minute hand point 20 minutes before the hour hand pointed to the 3?
If the clock spun backwards, then 20 minutes before any hour the minute hand would point to 4 assuming that the numbers remained painted on the dial as they are now.
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To solve this problem, let's break it down step by step.
1. First, let's determine where the hour hand would be when it points to the 3. In a round clock, the hour hand moves a total of 360 degrees over 12 hours, so it moves 30 degrees per hour.
To find where the hour hand would be, we need to calculate the number of hours from 12 to 3 and multiply it by the degree it moves in one hour: 3 - 12 = -9 hours. Since the motion is counter-clockwise, we use a negative sign. Therefore, the hour hand would be at -9 * 30 = -270 degrees.
2. Next, we need to calculate where the minute hand would be 20 minutes before the hour hand points to the 3. The minute hand moves a total of 360 degrees over 60 minutes, so it moves 6 degrees per minute.
To find where the minute hand would be, we calculate the number of minutes before the hour hand reaches the 3: 20 minutes.
Since the minute hand moves counter-clockwise, we need to subtract the number of degrees it moves from the degree position of the hour hand. So, -270 degrees - (20 minutes * 6 degrees/minute) = -270 degrees - 120 degrees = -390 degrees.
3. However, the minute hand can't point to -390 degrees on a clock because it only covers a 360-degree range from 0 to 360 degrees. So, we need to normalize the position of the minute hand.
To do that, we divide -390 degrees by 360 degrees: -390 degrees / 360 degrees = -1 remainder -30 degrees.
Since the minute hand can't point to negative degrees, we add 360 degrees to -30 degrees to get a positive equivalent angle: -30 degrees + 360 degrees = 330 degrees.
Therefore, if the hands of a round clock moved counter-clockwise, the minute hand would point to the number 11, because 330 degrees on the clock corresponds to the 11 o'clock position.