Ch. 3 - Radian Measure and The Unit Circle

Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo

Lial 12th Edition

Ch. 3 - Radian Measure and The Unit Circle

Problema 43

Capitolo 4, Problema 43

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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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 \(\times 1\) hour = \(4.5 \times 60\) 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 \(C = 2 \pi \times 3 = 6 \pi\) inches.

Multiply the circumference by the number of rotations to find the total distance traveled: \(\text{Distance} = 6 \pi \times 4.5\). Leave the answer as a multiple of \(\pi\) as requested.

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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 represents the distance traveled along the circumference. For full rotations, the total distance is the circumference, 2πr, where r is the radius.

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Relationship Between Time and Angle in Clock Hands

The minute hand completes one full rotation (360° or 2π radians) every 60 minutes. To find the angle swept over a given time, convert the time into minutes and calculate the fraction of the full rotation corresponding to that duration.

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Converting Hours to Minutes and Using Multiples of π

Since the problem asks for answers as multiples of π, keep π symbolic in calculations. Convert hours to minutes to determine the number of rotations, then multiply by the circumference (2πr) to find the total distance traveled by the tip.

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