The minute hand of a clock is 12 cm long, and the hour hand is 10cm long. Determine the distance between the tips of the hands at 9:30pm, to the nearest mm.
The answer is 17.5 cm I need to work on how to get to that answer!
at 9:30 the hour hand has gone clockwise 9.5/12 * 360 = 285° from the top
The minute hand is at 180°
So, the angle between them is 105°. You want d where
d^2 = 12^2 + 10^2 - 2*12*10*cos 105°
d^2 = 306.1
d = 17.5
Just a straightforward application of the law of cosines.
To find the distance between the tips of the hands at 9:30 pm, we need to calculate the length of each hand's arc from the 12 o'clock position to the given time, and then find the straight line distance between their tips.
First, let's calculate the angle covered by each hand:
- The hour hand moves 30 degrees per hour, and since it's 9:30 pm, it has covered 9.5 hours or 285 degrees (30 × 9.5).
- The minute hand moves 360 degrees per hour, and since it's 9:30 pm, it has covered 9.5 hours or 3420 degrees (360 × 9.5).
Now, let's find the length of each hand's arc:
- The minute hand has a length of 12 cm and has covered 3420 degrees. Since the circumference of a circle is 2πr, the length of the arc covered by the minute hand is (3420/360) × (2π × 12) cm.
- The hour hand has a length of 10 cm and has covered 285 degrees. Using the same formula, the length of the arc covered by the hour hand is (285/360) × (2π × 10) cm.
Next, let's find the position of the tips of each hand:
- The tip of the minute hand is at the end of its arc, which is the distance calculated above.
- The tip of the hour hand is also at the end of its arc, which is the distance calculated above.
Finally, we can find the straight line distance between the tips of the hands using the distance formula (Pythagorean theorem):
- The x-coordinate of the tip of the minute hand is the length of its arc, and the y-coordinate is 0.
- The x-coordinate of the tip of the hour hand is the length of its arc, and the y-coordinate is 0.
- Using the distance formula, we can find the distance between these two points.
By following these steps and performing the calculations, the distance between the tips of the hands at 9:30 pm should be approximately 17.5 cm.