For Exercises 57 and 58, refer to the following: The human body temperature normally fluctuates during the day. A person's body temperature can be predicted by the formula T 0.5 sin x + where x is the number of hours since 12 T = 99.1 midnight and T is in degrees Fahrenheit. 57. Body Temperature. According to this model, what is a person's temperature at 6:00 A.M.?
For Exercises 57 and 58, refer to the following: The human body temperature normally fluctuates during the day. A person's body temperature can be predicted by the formula T 0.5 sin x + where x is the number of hours since 12 T = 99.1 midnight and T is in degrees Fahrenheit. 57. Body Temperature. According to this model, what is a person's temperature at 6:00 A.M.?
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![### Predicting Body Temperature Using a Mathematical Model
For Exercises 57 and 58, refer to the following:
The human body temperature normally fluctuates during the day. A person’s body temperature can be predicted by the formula:
\[ T = 99.1 - 0.5 \sin \left( x + \frac{\pi}{12} \right) \]
where \( x \) is the number of hours since midnight and \( T \) is in degrees Fahrenheit.
#### 57. Body Temperature
**Question:** According to this model, what is a person’s temperature at 6:00 A.M.?
To find the answer, we need to substitute the appropriate value of \( x \) into the formula. Since 6:00 A.M. is 6 hours past midnight, we let \( x = 6 \).
\[ T = 99.1 - 0.5 \sin \left( 6 + \frac{\pi}{12} \right) \]
Using the value of \( \sin \left( 6 + \frac{\pi}{12} \right) \), we can compute the person's temperature.
### Detailed Explanation of the Formula and Concepts
1. **Equation Explanation:**
- \( T \) represents the body temperature in degrees Fahrenheit.
- \( x \) is the number of hours since midnight.
- \( \sin \) function is used to represent the cyclical nature of body temperature fluctuations during the day.
- \( 99.1 \) is the baseline body temperature around which fluctuations occur.
- \( 0.5 \) represents the amplitude of temperature fluctuations.
2. **Understanding Sinusoidal Functions:**
- Sinusoidal functions are used to model periodic phenomena. In this context, they capture the daily cycle of body temperature changes.
- The term \( \frac{\pi}{12} \) inside the sine function adjusts the phase of the sine wave, ensuring the function aligns correctly with the time of day.
By understanding these aspects, students can solve for body temperatures at various times and appreciate how mathematical models describe physiological processes.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb7c14283-9a82-48f7-90b1-831241c404f8%2Fd535011c-9a0a-47c5-8f45-279592081ab3%2F8e6g63_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Predicting Body Temperature Using a Mathematical Model
For Exercises 57 and 58, refer to the following:
The human body temperature normally fluctuates during the day. A person’s body temperature can be predicted by the formula:
\[ T = 99.1 - 0.5 \sin \left( x + \frac{\pi}{12} \right) \]
where \( x \) is the number of hours since midnight and \( T \) is in degrees Fahrenheit.
#### 57. Body Temperature
**Question:** According to this model, what is a person’s temperature at 6:00 A.M.?
To find the answer, we need to substitute the appropriate value of \( x \) into the formula. Since 6:00 A.M. is 6 hours past midnight, we let \( x = 6 \).
\[ T = 99.1 - 0.5 \sin \left( 6 + \frac{\pi}{12} \right) \]
Using the value of \( \sin \left( 6 + \frac{\pi}{12} \right) \), we can compute the person's temperature.
### Detailed Explanation of the Formula and Concepts
1. **Equation Explanation:**
- \( T \) represents the body temperature in degrees Fahrenheit.
- \( x \) is the number of hours since midnight.
- \( \sin \) function is used to represent the cyclical nature of body temperature fluctuations during the day.
- \( 99.1 \) is the baseline body temperature around which fluctuations occur.
- \( 0.5 \) represents the amplitude of temperature fluctuations.
2. **Understanding Sinusoidal Functions:**
- Sinusoidal functions are used to model periodic phenomena. In this context, they capture the daily cycle of body temperature changes.
- The term \( \frac{\pi}{12} \) inside the sine function adjusts the phase of the sine wave, ensuring the function aligns correctly with the time of day.
By understanding these aspects, students can solve for body temperatures at various times and appreciate how mathematical models describe physiological processes.
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