Outside temperature over a day can be modelled as a sinusoidal function. Suppose you know the temperature varies between 49 and 71 degrees during the day and the average daily temperature first occurs at 12 AM. Assuming t is the number of hours since midnight, find an equation for the temperature, Din terms of t
Outside temperature over a day can be modelled as a sinusoidal function. Suppose you know the temperature varies between 49 and 71 degrees during the day and the average daily temperature first occurs at 12 AM. Assuming t is the number of hours since midnight, find an equation for the temperature, Din terms of t
Chapter6: Exponential And Logarithmic Functions
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![### Modeling Daily Temperature as a Sinusoidal Function
The outside temperature over the course of a day can be effectively modeled using a sinusoidal function. Given the following information:
- The temperature varies between 49 degrees and 71 degrees throughout the day.
- The average daily temperature, which is the midpoint of the temperature range, first occurs at 12 AM (midnight).
- \( t \) represents the number of hours since midnight.
The objective is to find an equation for the temperature, \( D \), as a function of time \( t \).
### Steps to Formulate the Equation
1. **Identify the Midline (Average Temperature):**
The midline or average temperature \( D_{avg} \) is calculated as the midpoint of the minimum and maximum temperatures:
\[
D_{avg} = \frac{49 + 71}{2} = 60
\]
2. **Determine the Amplitude:**
The amplitude, which represents the difference between the maximum (or minimum) temperature and the average temperature, is given by:
\[
\text{Amplitude} = \frac{71 - 49}{2} = 11
\]
3. **Establish the Period:**
Since the temperature cycle repeats every 24 hours, the period \( T \) is 24 hours. The period of the sinusoidal function is related to the coefficient \( B \) in the following way:
\[
T = \frac{2\pi}{B} \quad \Rightarrow \quad B = \frac{2\pi}{24} = \frac{\pi}{12}
\]
4. **Find the Phase Shift:**
The phase shift is determined by the fact that the average temperature first occurs at \( t = 0 \). Therefore, there is no horizontal shift required for this function.
### Final Equation
Combining these components, we arrive at the equation for \( D(t) \):
\[
D(t) = D_{avg} + \text{Amplitude} \cdot \cos\left(Bt\right)
\]
Substituting the known values:
\[
D(t) = 60 + 11 \cdot \cos\left(\frac{\pi}{12}t\right)
\]
Thus, the equation modeling the temperature over the day as a function of time is:
\[
D(t) = 60](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb98bdb5b-53bd-47c2-bd74-197e9ff57506%2F1d60d8a5-054b-4c9c-aee5-bad88032836a%2Futub5sp_processed.png&w=3840&q=75)
Transcribed Image Text:### Modeling Daily Temperature as a Sinusoidal Function
The outside temperature over the course of a day can be effectively modeled using a sinusoidal function. Given the following information:
- The temperature varies between 49 degrees and 71 degrees throughout the day.
- The average daily temperature, which is the midpoint of the temperature range, first occurs at 12 AM (midnight).
- \( t \) represents the number of hours since midnight.
The objective is to find an equation for the temperature, \( D \), as a function of time \( t \).
### Steps to Formulate the Equation
1. **Identify the Midline (Average Temperature):**
The midline or average temperature \( D_{avg} \) is calculated as the midpoint of the minimum and maximum temperatures:
\[
D_{avg} = \frac{49 + 71}{2} = 60
\]
2. **Determine the Amplitude:**
The amplitude, which represents the difference between the maximum (or minimum) temperature and the average temperature, is given by:
\[
\text{Amplitude} = \frac{71 - 49}{2} = 11
\]
3. **Establish the Period:**
Since the temperature cycle repeats every 24 hours, the period \( T \) is 24 hours. The period of the sinusoidal function is related to the coefficient \( B \) in the following way:
\[
T = \frac{2\pi}{B} \quad \Rightarrow \quad B = \frac{2\pi}{24} = \frac{\pi}{12}
\]
4. **Find the Phase Shift:**
The phase shift is determined by the fact that the average temperature first occurs at \( t = 0 \). Therefore, there is no horizontal shift required for this function.
### Final Equation
Combining these components, we arrive at the equation for \( D(t) \):
\[
D(t) = D_{avg} + \text{Amplitude} \cdot \cos\left(Bt\right)
\]
Substituting the known values:
\[
D(t) = 60 + 11 \cdot \cos\left(\frac{\pi}{12}t\right)
\]
Thus, the equation modeling the temperature over the day as a function of time is:
\[
D(t) = 60
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