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 . 30 min

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Identify the radius of the circular path traveled by the tip of the minute hand. Since the tip is 3 inches from the center, the radius \(r\) is 3 inches.

Recall that the distance traveled by a point moving along a circular path is the length of the arc it covers. The formula for arc length \(s\) is \(s = r \theta\), where \(\theta\) is the central angle in radians.

Determine the fraction of the full circle the minute hand moves in 30 minutes. Since the minute hand completes one full rotation (360 degrees or \(2\pi\) radians) in 60 minutes, in 30 minutes it moves half the circle, so \(\theta = \pi\) radians.

Calculate the arc length using the formula \(s = r \theta\). Substitute \(r = 3\) inches and \(\theta = \pi\) radians to express the distance traveled as a multiple of \(\pi\).

Write the final expression for the distance traveled by the tip of the minute hand in 30 minutes as \(3\pi\) inches, leaving the answer in terms of \(\pi\) as requested.

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주요 개념

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Arc Length Formula

The arc length of a circle segment is calculated by multiplying the radius by the central angle in radians. It is given by the formula s = rθ, where s is the arc length, r is the radius, and θ is the angle in radians. This formula helps find the distance traveled along the circular path.

Conversion Between Degrees and Radians

Angles can be measured in degrees or radians, where 360° equals 2π radians. To use the arc length formula, convert degrees to radians using the relation radians = (π/180) × degrees. This conversion is essential for applying trigonometric formulas correctly.

Minute Hand Movement and Angular Displacement

The minute hand completes a full 360° rotation every 60 minutes, so its angular speed is 6° per minute. For a given time, the angular displacement is found by multiplying the time by this rate. Understanding this helps determine the angle subtended by the minute hand's movement.