Answer :
Let's walk through the solution step-by-step:
1. Domain Restriction (1st Year and Spawning Seasons):
- (a) First Year: Since [tex]\( t \)[/tex] is in months after December 2019, the first year would be from [tex]\( t = 0 \)[/tex] to [tex]\( t = 12 \)[/tex]. Therefore, the appropriate domain to restrict [tex]\( f \)[/tex] to is [tex]\( [0, 12] \)[/tex].
- (b) Alaskan Coho salmon (July-November, First Year): These months correspond to [tex]\( t = 7 \)[/tex] to [tex]\( t = 11 \)[/tex] assuming January is March 2020. Hence, the domain is [tex]\( [7, 11] \)[/tex].
- (c) Alaskan Coho salmon (July-November, First Two Years): This includes two spawning periods:
- First Year: [tex]\( t = 7 \)[/tex] to [tex]\( t = 11 \)[/tex]
- Second Year: [tex]\( t = 19 \)[/tex] to [tex]\( t = 23 \)[/tex] (One year later)
- Therefore, the domain is the union: [tex]\( [7, 11] \cup [19, 23] \)[/tex].
2. Local Minimums and Maximums (Critical Values Analysis):
- Increasing and Decreasing Temperatures:
- In Alaska, temperatures typically rise from January to June and fall from July to January.
- For the given critical values, [tex]\( t = 5.743 \)[/tex] is in the first half of the cycle (March), suggesting it's likely a minimum.
- [tex]\( t = 11.97 \)[/tex], occurring around July, when temperatures tend to decrease again, is likely a maximum.
3. Date Calculation for [tex]\( t = 62.25 \)[/tex]:
- Months Since December 2019:
- [tex]\( t = 62.25 \)[/tex] months after December 2019 corresponds to June 2025.
- To get the specific day, consider 0.25 of a month (approximately 7 days assuming 30 days per month).
- Therefore, the date would be June 7, 2025.
4. Interpretation for Alaskan Coho Salmon Spawning:
- Relevant Critical Points: Since Coho salmon spawn from July to November, we look for critical values within those periods:
- [tex]\( t = 68.56 \)[/tex] and [tex]\( t = 74.81 \)[/tex] appear during the July-November timespan two years after December 2019.
5. Model Suitability for California Coho Salmon:
- Comparison with California Data:
- The model [tex]\( f \)[/tex] was based on specific Alaskan conditions, and using it for California Coho salmon might not be suitable.
- In California, at a mean water temperature of [tex]\( 10.5^\circ C \)[/tex], eggs hatch in 38 days, which doesn't directly align with the model's predictions for Alaska.
In conclusion, our analysis respects the local seasonal patterns and critical times for Alaskan Coho spawned hatching and compares them critically to another region—California, establishing the contextual accuracy and limitations of the given model.
1. Domain Restriction (1st Year and Spawning Seasons):
- (a) First Year: Since [tex]\( t \)[/tex] is in months after December 2019, the first year would be from [tex]\( t = 0 \)[/tex] to [tex]\( t = 12 \)[/tex]. Therefore, the appropriate domain to restrict [tex]\( f \)[/tex] to is [tex]\( [0, 12] \)[/tex].
- (b) Alaskan Coho salmon (July-November, First Year): These months correspond to [tex]\( t = 7 \)[/tex] to [tex]\( t = 11 \)[/tex] assuming January is March 2020. Hence, the domain is [tex]\( [7, 11] \)[/tex].
- (c) Alaskan Coho salmon (July-November, First Two Years): This includes two spawning periods:
- First Year: [tex]\( t = 7 \)[/tex] to [tex]\( t = 11 \)[/tex]
- Second Year: [tex]\( t = 19 \)[/tex] to [tex]\( t = 23 \)[/tex] (One year later)
- Therefore, the domain is the union: [tex]\( [7, 11] \cup [19, 23] \)[/tex].
2. Local Minimums and Maximums (Critical Values Analysis):
- Increasing and Decreasing Temperatures:
- In Alaska, temperatures typically rise from January to June and fall from July to January.
- For the given critical values, [tex]\( t = 5.743 \)[/tex] is in the first half of the cycle (March), suggesting it's likely a minimum.
- [tex]\( t = 11.97 \)[/tex], occurring around July, when temperatures tend to decrease again, is likely a maximum.
3. Date Calculation for [tex]\( t = 62.25 \)[/tex]:
- Months Since December 2019:
- [tex]\( t = 62.25 \)[/tex] months after December 2019 corresponds to June 2025.
- To get the specific day, consider 0.25 of a month (approximately 7 days assuming 30 days per month).
- Therefore, the date would be June 7, 2025.
4. Interpretation for Alaskan Coho Salmon Spawning:
- Relevant Critical Points: Since Coho salmon spawn from July to November, we look for critical values within those periods:
- [tex]\( t = 68.56 \)[/tex] and [tex]\( t = 74.81 \)[/tex] appear during the July-November timespan two years after December 2019.
5. Model Suitability for California Coho Salmon:
- Comparison with California Data:
- The model [tex]\( f \)[/tex] was based on specific Alaskan conditions, and using it for California Coho salmon might not be suitable.
- In California, at a mean water temperature of [tex]\( 10.5^\circ C \)[/tex], eggs hatch in 38 days, which doesn't directly align with the model's predictions for Alaska.
In conclusion, our analysis respects the local seasonal patterns and critical times for Alaskan Coho spawned hatching and compares them critically to another region—California, establishing the contextual accuracy and limitations of the given model.