The diameter of clock's face is 6 inches. Find the length of the minor arc formed by the hands of the clock at 4:00.
The length of the minor arc formed by the hands of the clock at 4:00 is 2π inches.
To find the length of the minor arc formed by the hands of the clock at 4:00, we need to determine the angle between the hour hand and the minute hand.
At 4:00, the minute hand points directly at the 12 on the clock face, while the hour hand is pointing at the 4.
Since the clock's face has a diameter of 6 inches, its radius is half that measurement, which is 3 inches.
Using the formula for calculating the length of an arc, we have:
Arc Length = (θ/360) * (2π * r)
where θ is the central angle in degrees, r is the radius of the circle.
To find θ, we need to calculate the angle between the hour hand and the minute hand.
At 4:00, the minute hand is on the 12, which is 360 degrees. The hour hand is pointing at the 4, which is one-third of the distance between 12 and 5, so it has covered 120 degrees.
Since the minute hand is pointing at the 12 and the hour hand is pointing at the 4, we subtract the angle of the hour hand from the angle of the minute hand:
θ = 360 - 120 = 240 degrees.
Now we can substitute the values into the formula:
Arc Length = (240/360) * (2π * 3)
= (2/3) * (2π * 3)
= (2/3) * (6π)
= 4π
So, the length of the minor arc formed by the hands of the clock at 4:00 is 4π inches.
To find the length of the minor arc formed by the hands of the clock at 4:00, we need to determine the angle that the hour and minute hands form at that time.
Step 1: Calculate the angle between the hour and minute hands.
At 4:00, the hour hand points to the 4 while the minute hand points to the 12. Since there are 12 hours on a clock, each hour mark represents 30 degrees (360 degrees divided by 12). Therefore, at 4:00, the hour hand is at an angle of 4 hours x 30 degrees/hour = 120 degrees.
The minute hand at 4:00 points directly to the 12, which is 0 degrees.
Step 2: Calculate the smaller angle between the hour and minute hands.
To find the smaller angle between the hour and minute hands, we subtract the smaller angle from the larger angle. In this case, the smaller angle is 0 degrees, and the larger angle is 120 degrees. Therefore, the smaller angle between the hour and minute hands at 4:00 is 120 degrees.
Step 3: Calculate the length of the minor arc.
The length of an arc can be calculated using the formula: length = (angle/360) x circumference.
In this case, the angle between the hour and minute hands is 120 degrees, and the circumference of the clock's face is the perimeter of a circle with a diameter of 6 inches. The formula for the circumference of a circle is circumference = π × diameter.
Substituting the values into the formula, we have:
length = (120/360) x (π × 6) = (1/3) x (3.14 × 6) = 2 π ≈ 6.28 inches.
Therefore, the length of the minor arc formed by the hands of the clock at 4:00 is approximately 6.28 inches.