Answer :
Sure, let's go through the process of modeling the hours of daylight with the information provided.
Given data:
- The number of hours of daylight on the summer solstice: [tex]\( y_{\text{summer solstice}} = 15.3 \)[/tex]
- The number of hours of daylight on the winter solstice: [tex]\( y_{\text{winter solstice}} = 9.1 \)[/tex]
- The period of the cycle: [tex]\( T = 365 \)[/tex] days
- The day of the year for the summer solstice: [tex]\( x_{\text{summer solstice}} = 172 \)[/tex]
- The day of the year for the winter solstice: [tex]\( x_{\text{winter solstice}} = 355 \)[/tex]
To create a function [tex]\( y = a \cos\left( \omega (x - \phi) \right) + D \)[/tex] that models the daylight hours, we need to determine:
1. The vertical shift ([tex]\( D \)[/tex]).
2. The amplitude ([tex]\( A \)[/tex]).
3. The angular frequency ([tex]\( \omega \)[/tex]).
4. The phase shift ([tex]\( \phi \)[/tex]).
Step-by-step solution:
1. Vertical Shift ([tex]\( D \)[/tex]):
The vertical shift is the average of the maximum and minimum values of the function.
[tex]\[ D = \frac{y_{\text{summer solstice}} + y_{\text{winter solstice}}}{2} \][/tex]
Given the values:
[tex]\[ D = \frac{15.3 + 9.1}{2} = 12.2 \][/tex]
2. Amplitude ([tex]\( A \)[/tex]):
The amplitude is half the difference between the maximum and minimum values of the function.
[tex]\[ A = \frac{y_{\text{summer solstice}} - y_{\text{winter solstice}}}{2} \][/tex]
Given the values:
[tex]\[ A = \frac{15.3 - 9.1}{2} = 3.1 \][/tex]
3. Angular Frequency ([tex]\( \omega \)[/tex]):
The angular frequency is related to the period of the function.
[tex]\[ \omega = \frac{2\pi}{T} \][/tex]
Given the period [tex]\( T = 365 \)[/tex] days:
[tex]\[ \omega = \frac{2\pi}{365} \approx 0.017214 \][/tex]
4. Phase Shift ([tex]\( \phi \)[/tex]):
The maximum value occurs at [tex]\( x = 172 \)[/tex] (the summer solstice). In a cosine function, this corresponds to [tex]\( \cos(0) = 1 \)[/tex], so we need to solve for [tex]\( \phi \)[/tex] when [tex]\( x = 172 \)[/tex].
[tex]\[ \omega (x_{\text{summer solstice}} - \phi) = 0 \][/tex]
Therefore:
[tex]\[ 0.017214 \times (172 - \phi) = 0 \][/tex]
Hence:
[tex]\[ \phi = 172 \][/tex]
Putting it all together, our function [tex]\( y = A \cos(\omega (x - \phi)) + D \)[/tex] becomes:
[tex]\[ y = 3.1 \cos \left( \frac{2\pi}{365} (x - 172) \right) + 12.2 \][/tex]
Given data:
- The number of hours of daylight on the summer solstice: [tex]\( y_{\text{summer solstice}} = 15.3 \)[/tex]
- The number of hours of daylight on the winter solstice: [tex]\( y_{\text{winter solstice}} = 9.1 \)[/tex]
- The period of the cycle: [tex]\( T = 365 \)[/tex] days
- The day of the year for the summer solstice: [tex]\( x_{\text{summer solstice}} = 172 \)[/tex]
- The day of the year for the winter solstice: [tex]\( x_{\text{winter solstice}} = 355 \)[/tex]
To create a function [tex]\( y = a \cos\left( \omega (x - \phi) \right) + D \)[/tex] that models the daylight hours, we need to determine:
1. The vertical shift ([tex]\( D \)[/tex]).
2. The amplitude ([tex]\( A \)[/tex]).
3. The angular frequency ([tex]\( \omega \)[/tex]).
4. The phase shift ([tex]\( \phi \)[/tex]).
Step-by-step solution:
1. Vertical Shift ([tex]\( D \)[/tex]):
The vertical shift is the average of the maximum and minimum values of the function.
[tex]\[ D = \frac{y_{\text{summer solstice}} + y_{\text{winter solstice}}}{2} \][/tex]
Given the values:
[tex]\[ D = \frac{15.3 + 9.1}{2} = 12.2 \][/tex]
2. Amplitude ([tex]\( A \)[/tex]):
The amplitude is half the difference between the maximum and minimum values of the function.
[tex]\[ A = \frac{y_{\text{summer solstice}} - y_{\text{winter solstice}}}{2} \][/tex]
Given the values:
[tex]\[ A = \frac{15.3 - 9.1}{2} = 3.1 \][/tex]
3. Angular Frequency ([tex]\( \omega \)[/tex]):
The angular frequency is related to the period of the function.
[tex]\[ \omega = \frac{2\pi}{T} \][/tex]
Given the period [tex]\( T = 365 \)[/tex] days:
[tex]\[ \omega = \frac{2\pi}{365} \approx 0.017214 \][/tex]
4. Phase Shift ([tex]\( \phi \)[/tex]):
The maximum value occurs at [tex]\( x = 172 \)[/tex] (the summer solstice). In a cosine function, this corresponds to [tex]\( \cos(0) = 1 \)[/tex], so we need to solve for [tex]\( \phi \)[/tex] when [tex]\( x = 172 \)[/tex].
[tex]\[ \omega (x_{\text{summer solstice}} - \phi) = 0 \][/tex]
Therefore:
[tex]\[ 0.017214 \times (172 - \phi) = 0 \][/tex]
Hence:
[tex]\[ \phi = 172 \][/tex]
Putting it all together, our function [tex]\( y = A \cos(\omega (x - \phi)) + D \)[/tex] becomes:
[tex]\[ y = 3.1 \cos \left( \frac{2\pi}{365} (x - 172) \right) + 12.2 \][/tex]
