The minute hand of a clock is 8 inches long. How far does the tip of the minute hand move in 40 minutes? Round to two decimal OA. 34.74 inches O B. 31.77 inches OC. 33.51 inches O D. 36.02 inches
The minute hand of a clock is 8 inches long. How far does the tip of the minute hand move in 40 minutes? Round to two decimal OA. 34.74 inches O B. 31.77 inches OC. 33.51 inches O D. 36.02 inches
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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Question
![**Question:**
The minute hand of a clock is 8 inches long. How far does the tip of the minute hand move in 40 minutes? Round to two decimal places.
**Options:**
A. 34.74 inches
B. 31.77 inches
C. 33.51 inches
D. 36.02 inches
**Explanation:**
To solve this problem, we need to determine the length of the arc that the tip of the minute hand travels in 40 minutes. The length of the arc (s) can be calculated using the formula:
\[ s = r \cdot \theta \]
where:
- \( r \) is the radius of the circle (which is the length of the minute hand),
- \( \theta \) is the central angle in radians.
First, we need to convert the time into the central angle. The clock is divided into 60 minutes per full revolution, corresponding to \( 360^\circ \). Therefore:
\[ \text{Angle per minute} = \frac{360^\circ}{60} = 6^\circ \]
For 40 minutes:
\[ \text{Central angle} (\theta) = 40 \cdot 6^\circ = 240^\circ \]
Now, we convert degrees to radians since the arc length formula requires the angle in radians:
\[ \theta_{\text{radians}} = 240^\circ \times \frac{\pi}{180^\circ} = \frac{4\pi}{3} \]
Using the formula \( s = r \cdot \theta \):
\[ s = 8 \text{ inches} \cdot \frac{4\pi}{3} \]
\[ s = 8 \cdot \frac{4\pi}{3} \approx 33.51 \text{ inches} \]
Therefore, the correct answer is:
**Option C: 33.51 inches.**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad727bdc-baba-4947-a2e3-e934fc451c32%2Fda3cb81c-a48b-4851-82fa-399c5c7697e3%2F4dbhjdo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question:**
The minute hand of a clock is 8 inches long. How far does the tip of the minute hand move in 40 minutes? Round to two decimal places.
**Options:**
A. 34.74 inches
B. 31.77 inches
C. 33.51 inches
D. 36.02 inches
**Explanation:**
To solve this problem, we need to determine the length of the arc that the tip of the minute hand travels in 40 minutes. The length of the arc (s) can be calculated using the formula:
\[ s = r \cdot \theta \]
where:
- \( r \) is the radius of the circle (which is the length of the minute hand),
- \( \theta \) is the central angle in radians.
First, we need to convert the time into the central angle. The clock is divided into 60 minutes per full revolution, corresponding to \( 360^\circ \). Therefore:
\[ \text{Angle per minute} = \frac{360^\circ}{60} = 6^\circ \]
For 40 minutes:
\[ \text{Central angle} (\theta) = 40 \cdot 6^\circ = 240^\circ \]
Now, we convert degrees to radians since the arc length formula requires the angle in radians:
\[ \theta_{\text{radians}} = 240^\circ \times \frac{\pi}{180^\circ} = \frac{4\pi}{3} \]
Using the formula \( s = r \cdot \theta \):
\[ s = 8 \text{ inches} \cdot \frac{4\pi}{3} \]
\[ s = 8 \cdot \frac{4\pi}{3} \approx 33.51 \text{ inches} \]
Therefore, the correct answer is:
**Option C: 33.51 inches.**
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