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Ch. 3 - Radian Measure and The Unit Circle
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Lial 12th EditionCh. 3 - Radian Measure and The Unit Circle문제 43
4장, 문제 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

검증된 단계별 안내
1
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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
3m

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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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가이드 코스
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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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