The number of hours of daylight in Boston is given by f(x) = 3 sin( 2 (x-79)) +12, where x is the number of days after January 1. What is the most 365 amount of daylight that Boston will experience? O 9 hours 3 hours 15 hours 12 hours
The number of hours of daylight in Boston is given by f(x) = 3 sin( 2 (x-79)) +12, where x is the number of days after January 1. What is the most 365 amount of daylight that Boston will experience? O 9 hours 3 hours 15 hours 12 hours
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 Daylight Variation in Boston
**Problem Statement:**
The number of hours of daylight in Boston is given by the function:
\[ f(x) = 3\sin\left(\frac{2\pi}{365}(x - 79)\right) + 12 \]
where \( x \) is the number of days after January 1. What is the most amount of daylight that Boston will experience?
**Options:**
1. 9 hours
2. 3 hours
3. 15 hours
4. 12 hours
### Explanation:
This mathematical model describes how the number of daylight hours in Boston changes throughout the year. The function is a sinusoidal model, which is common for representing periodic phenomena like daylight hours.
**Breakdown of the Function:**
- **Amplitude:** The coefficient 3 in \( 3\sin(...) \) indicates the amplitude of the sine wave. This tells us that the daylight hours vary by ±3 hours from the midline over the course of the year.
- **Midline:** The constant 12 in the function \( +12 \) indicates the average or midline number of daylight hours around which the sine wave oscillates.
- **Period:** The value \( \frac{2\pi}{365} \) adjusts the period of the sine wave to align with the annual cycle of 365 days, which matches the length of a year.
- **Phase Shift:** The \( (x - 79) \) inside the sine function shifts the sine wave horizontally, accounting for the timing of the Earth's orbit and the seasonal changes in daylight.
### Calculation:
1. **Maximum Daylight:** To find the maximum amount of daylight hours, we calculate the maximum value of the sine function, which is 1 for \( \sin \theta \).
Therefore, the maximum value of \( 3\sin\left(\frac{2\pi}{365}(x - 79)\right) \) is 3.
Adding this to the midline value of 12 gives us:
\[
12 + 3 = 15 \text{ hours}
\]
### Answer:
The correct answer is **15 hours**. This is the maximum number of daylight hours that Boston will experience.
### Visual Representation:
- **Graph of the Function:** In a typical educational setting, you might provide a graph showing the number of daylight hours (y-axis) versus the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbb4c9257-c3a4-42ab-bf0e-0de9c4c5a001%2Fd2449126-5bbb-4b84-b788-75d97c3d019e%2Fni75a2e_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Daylight Variation in Boston
**Problem Statement:**
The number of hours of daylight in Boston is given by the function:
\[ f(x) = 3\sin\left(\frac{2\pi}{365}(x - 79)\right) + 12 \]
where \( x \) is the number of days after January 1. What is the most amount of daylight that Boston will experience?
**Options:**
1. 9 hours
2. 3 hours
3. 15 hours
4. 12 hours
### Explanation:
This mathematical model describes how the number of daylight hours in Boston changes throughout the year. The function is a sinusoidal model, which is common for representing periodic phenomena like daylight hours.
**Breakdown of the Function:**
- **Amplitude:** The coefficient 3 in \( 3\sin(...) \) indicates the amplitude of the sine wave. This tells us that the daylight hours vary by ±3 hours from the midline over the course of the year.
- **Midline:** The constant 12 in the function \( +12 \) indicates the average or midline number of daylight hours around which the sine wave oscillates.
- **Period:** The value \( \frac{2\pi}{365} \) adjusts the period of the sine wave to align with the annual cycle of 365 days, which matches the length of a year.
- **Phase Shift:** The \( (x - 79) \) inside the sine function shifts the sine wave horizontally, accounting for the timing of the Earth's orbit and the seasonal changes in daylight.
### Calculation:
1. **Maximum Daylight:** To find the maximum amount of daylight hours, we calculate the maximum value of the sine function, which is 1 for \( \sin \theta \).
Therefore, the maximum value of \( 3\sin\left(\frac{2\pi}{365}(x - 79)\right) \) is 3.
Adding this to the midline value of 12 gives us:
\[
12 + 3 = 15 \text{ hours}
\]
### Answer:
The correct answer is **15 hours**. This is the maximum number of daylight hours that Boston will experience.
### Visual Representation:
- **Graph of the Function:** In a typical educational setting, you might provide a graph showing the number of daylight hours (y-axis) versus the
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