Question

Distance Traveled by a Minute Hand Suppose the tip of the minute hand of a clock is 3 in. from the center of the clock. For each duration, determine the distance traveled by the tip of the minute hand. Leave answers as multiples of π . 4.5 hr

Accepted Answer

Identify the radius of the circular path traced by the tip of the minute hand. Here, the radius $$r is $$given as 3 inches. Recall that the distance traveled by the tip of the minute hand corresponds to the length of the arc it sweeps along the circle. The total distance traveled depends on the number of full rotations made in the given time. Calculate the number of full rotations the minute hand makes in 4.5 hours. Since the minute hand completes one full rotation every 60 minutes (1 hour), the number of rotations is $$4.5$$ rotations per hour $$	imes 1$$ hour = $$4.5 	imes 60$$ minutes divided by 60 minutes per rotation, which simplifies to $$4.5$$ rotations. Use the formula for the circumference of a circle, which is the distance traveled in one full rotation: $$C = 2 \pi r. $$Substitute $$r = 3$$ inches to get $$C = 2 \pi 	imes 3 = 6 \pi$$ inches. Multiply the circumference by the number of rotations to find the total distance traveled: $$	ext{Distance} = 6 \pi 	imes 4.5. $$Leave the answer as a multiple of $$\pi as $$requested.

Distance Traveled by a Minute Hand Suppose the tip of the minute hand of a clock is 3 in. from the center of the clock. For each duration, determine the distance traveled by the tip of the minute hand. Leave answers as multiples of π . 4.5 hr

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Key Concepts

Arc Length Formula

Relationship Between Time and Angle in Clock Hands

Converting Hours to Minutes and Using Multiples of π