A clock with an hour hand that is 18 inches long is hanging on a wall. At noon, the vertical distance between the tip of the hour hand and the ceiling is 25 inches. At 3 P.M., the distance is 43 inches; at 6 P.M., 61 inches; at 9 P.M., 43 inches; and at midnight the distance is again 25 inches. If f (x) represents he distance between the tip of the hour hand and the ceiling x hours after noon, write the function that epresents this situation in the form of f(x) = acos(bx) + d. о f(x) = -18 cos Of(x)=-18 cos(x) + 43 +43 Of(x)=-43 cos (*) +18 Of(x)=-43 cos (x) + +18
A clock with an hour hand that is 18 inches long is hanging on a wall. At noon, the vertical distance between the tip of the hour hand and the ceiling is 25 inches. At 3 P.M., the distance is 43 inches; at 6 P.M., 61 inches; at 9 P.M., 43 inches; and at midnight the distance is again 25 inches. If f (x) represents he distance between the tip of the hour hand and the ceiling x hours after noon, write the function that epresents this situation in the form of f(x) = acos(bx) + d. о f(x) = -18 cos Of(x)=-18 cos(x) + 43 +43 Of(x)=-43 cos (*) +18 Of(x)=-43 cos (x) + +18
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Understanding Periodic Functions: A Practical Example with a Clock's Hour Hand
Consider a clock with an hour hand that is 18 inches long, mounted vertically on a wall. At different times of the day, there will be varying distances between the tip of the hour hand and the ceiling. This problem explores how to model such a situation using a cosine function.
#### Problem Description:
- **At noon**: The vertical distance between the tip of the hour hand and the ceiling is **25 inches**.
- **At 3 PM**: The distance is **43 inches**.
- **At 6 PM**: The distance is **61 inches**.
- **At 9 PM**: The distance is **43 inches** again.
- **At midnight**: The distance returns to **25 inches**.
The goal is to represent the distance between the tip of the hour hand and the ceiling, \( f(x) \), as a function of time, \( x \), where \( x \) is the number of hours after noon. The function is to be expressed in the form:
\[ f(x) = a \cos(bx) + d \]
#### Answer Choices to Consider:
1. \[ f(x) = -18 \cos\left(\frac{\pi}{12} x\right) + 43 \]
2. \[ f(x) = -18 \cos\left(\frac{\pi}{6} x\right) + 43 \]
3. \[ f(x) = -43 \cos\left(\frac{\pi}{12} x\right) + 18 \]
4. \[ f(x) = -43 \cos\left(\frac{\pi}{6} x\right) + 18 \]
### Explanation:
To solve for \( f(x) \), we need to determine the parameters \( a \), \( b \), and \( d \) by considering the behavior of the hour hand.
1. **Amplitude (a)**:
- The maximum deviation from the average distance (centerline) of 43 inches is 18 inches. Therefore, \( a = -18 \) (the negative sign reflects the phase shift to match the starting point conditions).
2. **Period (P)**:
- The hour hand completes a full cycle every 12 hours. Hence, the period \( P \) is 12.
- Using the period to calculate \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff19158ce-506a-4a94-9506-84274a4b7439%2F05337413-6855-4b6e-ae6f-e3bd3ee797aa%2Fpmlu5dc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Periodic Functions: A Practical Example with a Clock's Hour Hand
Consider a clock with an hour hand that is 18 inches long, mounted vertically on a wall. At different times of the day, there will be varying distances between the tip of the hour hand and the ceiling. This problem explores how to model such a situation using a cosine function.
#### Problem Description:
- **At noon**: The vertical distance between the tip of the hour hand and the ceiling is **25 inches**.
- **At 3 PM**: The distance is **43 inches**.
- **At 6 PM**: The distance is **61 inches**.
- **At 9 PM**: The distance is **43 inches** again.
- **At midnight**: The distance returns to **25 inches**.
The goal is to represent the distance between the tip of the hour hand and the ceiling, \( f(x) \), as a function of time, \( x \), where \( x \) is the number of hours after noon. The function is to be expressed in the form:
\[ f(x) = a \cos(bx) + d \]
#### Answer Choices to Consider:
1. \[ f(x) = -18 \cos\left(\frac{\pi}{12} x\right) + 43 \]
2. \[ f(x) = -18 \cos\left(\frac{\pi}{6} x\right) + 43 \]
3. \[ f(x) = -43 \cos\left(\frac{\pi}{12} x\right) + 18 \]
4. \[ f(x) = -43 \cos\left(\frac{\pi}{6} x\right) + 18 \]
### Explanation:
To solve for \( f(x) \), we need to determine the parameters \( a \), \( b \), and \( d \) by considering the behavior of the hour hand.
1. **Amplitude (a)**:
- The maximum deviation from the average distance (centerline) of 43 inches is 18 inches. Therefore, \( a = -18 \) (the negative sign reflects the phase shift to match the starting point conditions).
2. **Period (P)**:
- The hour hand completes a full cycle every 12 hours. Hence, the period \( P \) is 12.
- Using the period to calculate \(
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