If the length of the hour and the minute hands of a clock are 4 cm and 6 cm, respectively, what is the distance in cm between the tips of the hands at two o'clock?
there are 30° per hour
at 2:00, the minute hand is at 90°-60°=30° (measuring from horizontal)
the hour hand is at 90-(60+30/6) = 25°
minute tip is at (6cos30°,6sin30°) = (5.2,3.0)
hour tip is at (4cos25°,4sin25°)=(3.6,1.7)
d^2 = 1.6^2 + 1.3^2
d = 2.1 cm
oops I was thinking 2:10
at 2:00, the hands are at (0,6) and (3.46,2)
d = 5.3 cm
To find the distance between the tips of the hour and minute hands at two o'clock, we need to calculate the length of the hypotenuse of the right triangle formed by the hour hand, minute hand, and the center of the clock.
Since it is two o'clock, the hour hand is pointing directly at the 2 on the clock, while the minute hand is pointing at the 12.
First, let's calculate the length of the hour hand path from the center of the clock to the 2. We can think of this as the shorter leg of a right triangle.
Using the Pythagorean theorem, we can calculate the length of this leg.
The length of the shorter leg = length of the hour hand = 4 cm
Next, we calculate the length of the minute hand path from the center of the clock to the 12. We can think of this as the longer leg of the right triangle.
The length of the longer leg = length of the minute hand = 6 cm
Now, we can use the Pythagorean theorem to find the length of the hypotenuse, which represents the distance between the tips of the hour and minute hands.
Hypotenuse = √(leg1^2 + leg2^2)
= √(4^2 + 6^2)
= √(16 + 36)
= √52
= 7.21 cm (approximately)
Therefore, the distance between the tips of the hands at two o'clock is approximately 7.21 cm.
To find the distance between the tips of the hour and minute hands at two o'clock, we need to understand their positions and then calculate the distance.
First, let's understand their positions:
At two o'clock, the hour hand points to the 2, while the minute hand points to the 12.
The minute hand is at the topmost position (12) of the clock, while the hour hand is at the 2.
Now, let's calculate the distance between the tips of the hands:
To calculate the distance, we need to consider the clockwise direction of the hands. At two o'clock, the angle between the two hands is 30 degrees. (360 degrees divided by 12 hours equals 30 degrees per hour.)
Now, we can use trigonometry to calculate the distance between the tips of the hands. We can use the cosine rule, which states that c^2 = a^2 + b^2 - 2ab*cos(C), where c is the distance we want to find, a and b are the lengths of the hour and minute hands, and C is the angle between them.
In this case, a = 4 cm, b = 6 cm, and C = 30 degrees.
Using the formula, we have:
c^2 = 4^2 + 6^2 - 2*4*6*cos(30)
c^2 = 16 + 36 - 48*cos(30)
c^2 = 52 - 48*(√3/2)
c^2 = 52 - 24√3
Finally, we can calculate c:
c = √(52 - 24√3)
So, the distance between the tips of the hour and minute hands at two o'clock is approximately √(52 - 24√3) cm.