Questions: The minute hand of a clock is 3 inches long. How far does the tip of the minute hand move in 35 minutes? How far does it move in 40 minutes? The tip of the minute hand moves inches in 35 minutes.

Solution Steps

To find out how far the tip of the minute hand moves, we need to calculate the arc length it travels. The minute hand completes a full circle (360 degrees) in 60 minutes. We can use the formula for the arc length of a circle, which is \( \text{Arc Length} = \theta \times r \), where \( \theta \) is the angle in radians and \( r \) is the radius (length of the minute hand).

1. Convert the time (in minutes) to the angle in radians.
2. Use the arc length formula to find the distance traveled.

To find the angle in radians for 35 minutes, we use the proportion of the time to a full circle: \[ \theta_{35} = \left( \frac{35}{60} \right) \times 2\pi = 3.6652 \text{ radians} \]

Similarly, for 40 minutes: \[ \theta_{40} = \left( \frac{40}{60} \right) \times 2\pi = 4.1888 \text{ radians} \]

Using the arc length formula \( \text{Arc Length} = \theta \times r \) with \( r = 3 \) inches: \[ \text{Arc Length}_{35} = 3.6652 \times 3 = 10.9956 \text{ inches} \]

Similarly, for 40 minutes: \[ \text{Arc Length}_{40} = 4.1888 \times 3 = 12.5664 \text{ inches} \]

Final Answer