College

In Week 7, we modeled the temperature of Kodiak, Alaska, ignoring the effects of climate change. To include climate change in our model, we use the temperature function:

[tex]\[ T(t) = \frac{5}{9} \ln (t+100)\left[3 \sin \left(\frac{1}{2} t+5\right)+10\right]-\frac{160}{9} \][/tex]

where [tex]\( t \)[/tex] is given in months after December 2019 and [tex]\( T \)[/tex] is the mean water temperature in degrees Celsius [tex]\(\left({ }^{\circ} C \right)\)[/tex] at the location where the eggs were left to hatch. Then, we compose [tex]\( T(t) \)[/tex] with the model:

[tex]\[ H(T) = \frac{e^{6.727}}{T+2.394} \][/tex]

where [tex]\( H \)[/tex] is the number of days it took for the eggs to hatch after spawning. Let:

[tex]\[ f(t) = (H \circ T)(t) \][/tex]

1.
(a) Suppose we are interested in comparing [tex]\( f \)[/tex] to the first year of data. What is the appropriate domain to restrict [tex]\( f \)[/tex] to?

(b) Suppose we are interested in comparing data specifically for Alaskan Coho salmon in the first year, which spawn from July to November. What is the appropriate domain to restrict [tex]\( f \)[/tex] to?

(c) Suppose we are interested in comparing data specifically for Alaskan Coho salmon in the first two years, which spawn from July to November. What is the appropriate domain to restrict [tex]\( f \)[/tex] to?

2. After an initial comparison of the data with [tex]\( f \)[/tex] for the first two years, we decide that [tex]\( f \)[/tex] fits our observations closely enough to warrant further investigation. Next, we want to check if [tex]\( f \)[/tex] matches the extremal behaviors observed in the data (i.e., the shortest and longest hatching periods).

Using a computer program, the critical values of [tex]\( f \)[/tex] are computed (up to four significant figures) as:

[tex]\[ t = 5.743, 11.97, 18.31, 24.54, 30.87, 37.11, \ldots \][/tex]

(a) In Alaska, daily temperatures typically increase from January to June and decrease from June to January. Determine whether [tex]\( t = 5.743 \)[/tex] and [tex]\( t = 11.97 \)[/tex] are local minimums or local maximums.

(b) Below is a table of the critical points and their corresponding outputs:

[tex]\[
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Months Since 2020 & 62.25 & 68.56 & 74.81 & 81.13 & 87.38 & 93.69 \\
\hline
Temperature ('C) & 2.013 & 19.25 & 2.303 & 19.77 & 2.573 & 20.26 \\
\hline
Days until Hatch & 189.4 & 38.56 & 177.7 & 37.66 & 168.0 & 36.85 \\
\hline
\end{tabular}
\][/tex]

[tex]\[
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Months Since 2020 & 99.95 & 106.3 & 112.5 & 118.8 & 125.1 & 131.4 \\
\hline
Temperature ('C) & 2.826 & 20.71 & 3.063 & 21.14 & 3.286 & 21.54 \\
\hline
Days until Hatch & 159.9 & 36.13 & 153.0 & 35.47 & 146.9 & 34.87 \\
\hline
\end{tabular}
\][/tex]

What is the day, month, and year for [tex]\( t = 62.25 \)[/tex]?

(c) Recall from 1(b) and 1(c) that Alaskan Coho salmon spawn from July to November. If we are investigating Alaskan Coho salmon, which critical points in the table have an interpretation in the model [tex]\( f \)[/tex]?

(d) According to the California Department of Fish and Wildlife, when the mean water temperature is [tex]\( 10.5^{\circ} C \)[/tex], California Coho salmon eggs hatch in 38 days. Is [tex]\( f \)[/tex] an effective model to study California Coho salmon in 2025?

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.