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Answer :
Let's explore the function Thaddeus is using to model daylight hours:
The function is [tex]\( D(t) = 2 \sin \left(\frac{\pi}{6} t\right) + 12 \)[/tex], where [tex]\( D \)[/tex] is the number of daylight hours, and [tex]\( t \)[/tex] is the time in months since January 1.
### Step-by-Step Explanation:
1. Understanding the Sinusoidal Function:
- The sine function, [tex]\(\sin(x)\)[/tex], oscillates between -1 and 1.
2. Amplitude and Vertical Shift:
- In the model, the sinusoidal part is [tex]\( 2 \sin \left(\frac{\pi}{6} t\right) \)[/tex]. The "2" in front of the sine function is the amplitude, indicating that the function oscillates 2 units above and below its central axis.
- The vertical shift is +12, which means the entire function is shifted upwards by 12 units.
3. Determining the Range of Daylight Hours:
- The minimum value of [tex]\(\sin(x)\)[/tex] is -1. So, the minimum value of [tex]\( 2 \sin \left(\frac{\pi}{6} t\right) \)[/tex] will be [tex]\( 2 \times (-1) = -2 \)[/tex].
- Adding the vertical shift of 12, the minimum number of daylight hours is [tex]\( 12 - 2 = 10 \)[/tex].
- The maximum value of [tex]\(\sin(x)\)[/tex] is 1. So, the maximum value of [tex]\( 2 \sin \left(\frac{\pi}{6} t\right) \)[/tex] will be [tex]\( 2 \times 1 = 2 \)[/tex].
- Adding the vertical shift of 12, the maximum number of daylight hours is [tex]\( 12 + 2 = 14 \)[/tex].
Therefore, over the course of a year, the least and greatest numbers of daylight hours are:
- Least: 10 hours
- Greatest: 14 hours
So, the correct choice is D. Least: 10 hours; greatest: 14 hours.
The function is [tex]\( D(t) = 2 \sin \left(\frac{\pi}{6} t\right) + 12 \)[/tex], where [tex]\( D \)[/tex] is the number of daylight hours, and [tex]\( t \)[/tex] is the time in months since January 1.
### Step-by-Step Explanation:
1. Understanding the Sinusoidal Function:
- The sine function, [tex]\(\sin(x)\)[/tex], oscillates between -1 and 1.
2. Amplitude and Vertical Shift:
- In the model, the sinusoidal part is [tex]\( 2 \sin \left(\frac{\pi}{6} t\right) \)[/tex]. The "2" in front of the sine function is the amplitude, indicating that the function oscillates 2 units above and below its central axis.
- The vertical shift is +12, which means the entire function is shifted upwards by 12 units.
3. Determining the Range of Daylight Hours:
- The minimum value of [tex]\(\sin(x)\)[/tex] is -1. So, the minimum value of [tex]\( 2 \sin \left(\frac{\pi}{6} t\right) \)[/tex] will be [tex]\( 2 \times (-1) = -2 \)[/tex].
- Adding the vertical shift of 12, the minimum number of daylight hours is [tex]\( 12 - 2 = 10 \)[/tex].
- The maximum value of [tex]\(\sin(x)\)[/tex] is 1. So, the maximum value of [tex]\( 2 \sin \left(\frac{\pi}{6} t\right) \)[/tex] will be [tex]\( 2 \times 1 = 2 \)[/tex].
- Adding the vertical shift of 12, the maximum number of daylight hours is [tex]\( 12 + 2 = 14 \)[/tex].
Therefore, over the course of a year, the least and greatest numbers of daylight hours are:
- Least: 10 hours
- Greatest: 14 hours
So, the correct choice is D. Least: 10 hours; greatest: 14 hours.
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