the hour hand of a clock is 5 inches long and the minute hand is 6 inches long. Determine the distance between the tips of the hand at 11:40 p.m.
please help i don't understand how to do this.
if you work it out this will help alot
The hard part is finding the angle between the two hands.
Each hour represents 30°
so 11:40 = 11 2/3 hrs of rotation
the hour had is at the 350° position and the minute hand is at the 240° position,
so the angle between the two hands is 350-240 = 110°
now use the cosine law:
x^2= 5^2 + 6^2 - 2(5)(6)cos 110°
= 61 - 60cos110
= 81.5212
x = √81.5212 = 9.03 inches.
To find the distance between the tips of the hour and minute hand at a given time, you need to calculate the angular displacement of each hand and then use that to find the distance.
First, let's calculate the angle covered by each hand:
1. Hour hand: The hour hand moves 360 degrees in 12 hours, so in one hour it moves 360/12 = 30 degrees. Since the time is 11:40 p.m., the hour hand has moved (11 + 40/60) * 30 = 355 degrees.
2. Minute hand: The minute hand moves 360 degrees in 60 minutes, so in one minute it moves 360/60 = 6 degrees. At 11:40 p.m., the minute hand has moved (40/60) * 6 = 4 degrees.
Now, to find the distance between the tips of the hands, you can treat the hour and minute hands as vectors. The x-component of each vector represents the horizontal displacement, and the y-component represents the vertical displacement.
Let's assume the clock is centered at the origin on the Cartesian plane. The x-coordinate of the hour hand tip is given by 5 * cos(angle of hour hand) and the y-coordinate is given by 5 * sin(angle of hour hand). Similarly, for the minute hand, the x-coordinate is 6 * cos(angle of minute hand) and the y-coordinate is 6 * sin(angle of minute hand).
Using these coordinates, you can calculate the distance between the tips of the hand using the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Let's substitute the values into the formula:
Distance = sqrt((6*cos(4) - 5*cos(355))^2 + (6*sin(4) - 5*sin(355))^2)
Evaluating this expression gives us the distance between the tips of the hand at 11:40 p.m.