Answer :
Sure, let's break down the question and answer it step-by-step.
### 1. Domain Restrictions
#### 1(a) First Year of Data
The first year after December 2019 would be covering the months from January 2020 to December 2020. Therefore, in terms of `t`, which counts months from December 2019, this would be from `t = 0` to `t = 12`.
Domain: [tex]\( t \in (0, 12) \)[/tex]
#### 1(b) Alaskan Coho Salmon Spawning (First Year)
Alaskan Coho salmon spawn from July to November, so we need to determine these months for the first year. From December 2019:
- July 2020 corresponds to [tex]\( t = 6 \)[/tex]
- November 2020 corresponds to [tex]\( t = 11 \)[/tex]
Domain: [tex]\( t \in (6, 11) \)[/tex]
#### 1(c) Alaskan Coho Salmon Spawning (First Two Years)
Similarly, for the first two years, we have:
- For the first year: [tex]\( t \in (6, 11) \)[/tex]
- For the second year:
- July 2021 is [tex]\( t = 6 + 12 = 18 \)[/tex]
- November 2021 is [tex]\( t = 11 + 12 = 23 \)[/tex]
Domain:
- First Year: [tex]\( t \in (6, 11) \)[/tex]
- Second Year: [tex]\( t \in (18, 23) \)[/tex]
### 2. Critical Values and Temperatures
#### 2(a) Local Minimums and Maximums
Given the general trend that temperatures in Alaska increase from January to June and decrease from June to January:
- At [tex]\( t = 5.743 \)[/tex] (approximately June), temperature is increasing to its peak, suggesting a local minimum.
- At [tex]\( t = 11.97 \)[/tex] (approximately December), temperature has been decreasing, suggesting a local maximum.
- Behavior:
- [tex]\( t = 5.743 \)[/tex]: Minimum
- [tex]\( t = 11.97 \)[/tex]: Maximum
#### 2(b) Date for [tex]\( t = 62.25 \)[/tex]
To find the specific date for [tex]\( t = 62.25 \)[/tex]:
- [tex]\( t = 62.25 \)[/tex] months after December 2019 corresponds to February 2025.
Date: 1st February 2025
#### 2(c) Interpretation for Alaskan Coho Salmon
For Alaskan Coho salmon, interested in July to November:
Critical points during these spawning months in the table include:
- From July 2025 (63 months) to November 2025 fall within these ranges in every second year:
- [tex]\( t = 62.25 \)[/tex]
- [tex]\( t = 74.81 \)[/tex]
- [tex]\( t = 87.38 \)[/tex]
- [tex]\( t = 99.95 \)[/tex]
- [tex]\( t = 112.5 \)[/tex]
- [tex]\( t = 125.1 \)[/tex]
These points correspond to spawning ranges over the years within the critical point data provided.
#### 2(d) Effectiveness for California Coho Salmon
- The mean water temperature for California Coho salmon is [tex]\( 10.5^\circ C \)[/tex] for egg hatching in 38 days.
- Given the complex environmental differences between Alaska and California, resulting in significant temperature variance, the model [tex]\( f(t) \)[/tex] developed based on Alaskan data is likely not effective for California Coho salmon for extrapolation into 2025.
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I hope this helps you understand how to determine the appropriate domains and interpretations based on the model provided! If you have further questions, feel free to ask.
### 1. Domain Restrictions
#### 1(a) First Year of Data
The first year after December 2019 would be covering the months from January 2020 to December 2020. Therefore, in terms of `t`, which counts months from December 2019, this would be from `t = 0` to `t = 12`.
Domain: [tex]\( t \in (0, 12) \)[/tex]
#### 1(b) Alaskan Coho Salmon Spawning (First Year)
Alaskan Coho salmon spawn from July to November, so we need to determine these months for the first year. From December 2019:
- July 2020 corresponds to [tex]\( t = 6 \)[/tex]
- November 2020 corresponds to [tex]\( t = 11 \)[/tex]
Domain: [tex]\( t \in (6, 11) \)[/tex]
#### 1(c) Alaskan Coho Salmon Spawning (First Two Years)
Similarly, for the first two years, we have:
- For the first year: [tex]\( t \in (6, 11) \)[/tex]
- For the second year:
- July 2021 is [tex]\( t = 6 + 12 = 18 \)[/tex]
- November 2021 is [tex]\( t = 11 + 12 = 23 \)[/tex]
Domain:
- First Year: [tex]\( t \in (6, 11) \)[/tex]
- Second Year: [tex]\( t \in (18, 23) \)[/tex]
### 2. Critical Values and Temperatures
#### 2(a) Local Minimums and Maximums
Given the general trend that temperatures in Alaska increase from January to June and decrease from June to January:
- At [tex]\( t = 5.743 \)[/tex] (approximately June), temperature is increasing to its peak, suggesting a local minimum.
- At [tex]\( t = 11.97 \)[/tex] (approximately December), temperature has been decreasing, suggesting a local maximum.
- Behavior:
- [tex]\( t = 5.743 \)[/tex]: Minimum
- [tex]\( t = 11.97 \)[/tex]: Maximum
#### 2(b) Date for [tex]\( t = 62.25 \)[/tex]
To find the specific date for [tex]\( t = 62.25 \)[/tex]:
- [tex]\( t = 62.25 \)[/tex] months after December 2019 corresponds to February 2025.
Date: 1st February 2025
#### 2(c) Interpretation for Alaskan Coho Salmon
For Alaskan Coho salmon, interested in July to November:
Critical points during these spawning months in the table include:
- From July 2025 (63 months) to November 2025 fall within these ranges in every second year:
- [tex]\( t = 62.25 \)[/tex]
- [tex]\( t = 74.81 \)[/tex]
- [tex]\( t = 87.38 \)[/tex]
- [tex]\( t = 99.95 \)[/tex]
- [tex]\( t = 112.5 \)[/tex]
- [tex]\( t = 125.1 \)[/tex]
These points correspond to spawning ranges over the years within the critical point data provided.
#### 2(d) Effectiveness for California Coho Salmon
- The mean water temperature for California Coho salmon is [tex]\( 10.5^\circ C \)[/tex] for egg hatching in 38 days.
- Given the complex environmental differences between Alaska and California, resulting in significant temperature variance, the model [tex]\( f(t) \)[/tex] developed based on Alaskan data is likely not effective for California Coho salmon for extrapolation into 2025.
---
I hope this helps you understand how to determine the appropriate domains and interpretations based on the model provided! If you have further questions, feel free to ask.