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.
At 12:00 the hands are together
After h hours, the hour hand has gone h/12 * 360 degrees
After m minutes, minute hand has gone m/60 * 360 degrees
9:30 = 9.5 hours
:30 = 30 minutes
So, the hour hand has gone 9.5/12*360 = 285° and the minute hand is at 180°
So, the angle between the hands is 105°
Finally, the distance d between the tips is
d^2 = 12^2 + 10^2 - 2*10*12 cos105°
To determine the distance between the tips of the clock hands at 9:30 PM, we need to calculate the difference in the positions of the hour and minute hands.
First, let's calculate the distance covered by the minute hand. In 30 minutes, the minute hand moves a distance equal to half the circumference of the clock. The circumference is given by 2πr, where r is the length of the minute hand.
Circumference = 2 * π * 12 cm
Now, let's calculate the distance covered by the hour hand. The hour hand moves as the minutes progress, covering a fraction of the circumference equal to the proportion of the time that has passed. At 9:30 PM, 9 hours and 30 minutes have passed.
We can calculate the angle covered by the hour hand in degrees as follows:
Angle = (hours * 30°) + (minutes * 0.5°)
Angle = (9 hours * 30°) + (30 minutes * 0.5°)
Next, we calculate the distance covered by the hour hand in cm using the formula:
Distance = (Angle / 360°) * Circumference of the hour hand
Finally, we calculate the distance between the tips of the hands by subtracting the distance covered by the hour hand from the distance covered by the minute hand.
Distance between the tips of the hands = Distance of minute hand - Distance of hour hand
Let's calculate the result:
Circumference = 2 * π * 12 cm = 24π cm ≈ 75.39 cm
Angle = (9 hours * 30°) + (30 minutes * 0.5°) = 270° + 15° = 285°
Distance of hour hand = (285° / 360°) * 10 cm = 2.85 cm
Distance between the tips of the hands = 75.39 cm - 2.85 cm
Converting the result from cm to mm, we get:
Distance between the tips of the hands ≈ 724.54 mm
Therefore, the distance between the tips of the hands at 9:30 PM is approximately 724.54 mm.