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Answer :
To find the least and greatest numbers of daylight hours over the course of a year using Thaddeus's model [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, follow these steps:
1. Identify the behavior of the sine function:
- The sine function, [tex]\( \sin(x) \)[/tex], oscillates between -1 and 1.
2. Determine the extreme values of the sine part of the function:
- The minimum value of [tex]\( \sin(x) \)[/tex] is -1.
- The maximum value of [tex]\( \sin(x) \)[/tex] is 1.
3. Calculate the corresponding values of [tex]\( D(t) \)[/tex]:
- When [tex]\( \sin\left(\frac{\pi}{6} t\right) = -1 \)[/tex]:
[tex]\[
D(t) = 2 \cdot (-1) + 12 = -2 + 12 = 10
\][/tex]
- When [tex]\( \sin\left(\frac{\pi}{6} t\right) = 1 \)[/tex]:
[tex]\[
D(t) = 2 \cdot 1 + 12 = 2 + 12 = 14
\][/tex]
4. Interpret the results:
- The least number of daylight hours is 10 hours.
- The greatest number of daylight hours is 14 hours.
Therefore, the least and greatest numbers of daylight hours over the course of a year are:
[tex]\[
\boxed{\text{Least: 10 hours; greatest: 14 hours}}
\][/tex]
Given these results, the correct answer is:
[tex]\[ \text{C. Least: 10 hours; greatest: 14 hours} \][/tex]
1. Identify the behavior of the sine function:
- The sine function, [tex]\( \sin(x) \)[/tex], oscillates between -1 and 1.
2. Determine the extreme values of the sine part of the function:
- The minimum value of [tex]\( \sin(x) \)[/tex] is -1.
- The maximum value of [tex]\( \sin(x) \)[/tex] is 1.
3. Calculate the corresponding values of [tex]\( D(t) \)[/tex]:
- When [tex]\( \sin\left(\frac{\pi}{6} t\right) = -1 \)[/tex]:
[tex]\[
D(t) = 2 \cdot (-1) + 12 = -2 + 12 = 10
\][/tex]
- When [tex]\( \sin\left(\frac{\pi}{6} t\right) = 1 \)[/tex]:
[tex]\[
D(t) = 2 \cdot 1 + 12 = 2 + 12 = 14
\][/tex]
4. Interpret the results:
- The least number of daylight hours is 10 hours.
- The greatest number of daylight hours is 14 hours.
Therefore, the least and greatest numbers of daylight hours over the course of a year are:
[tex]\[
\boxed{\text{Least: 10 hours; greatest: 14 hours}}
\][/tex]
Given these results, the correct answer is:
[tex]\[ \text{C. Least: 10 hours; greatest: 14 hours} \][/tex]
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