**Calculus I with Applications**

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

[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]\((^{\circ} C)\)[/tex] of the location where the eggs were left to hatch. We then 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 the eggs to hatch after spawning (and [tex]\( T \)[/tex] as above). 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 that 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 that 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 (in other words, 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 daily temperatures typically 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]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
\text{Months Since 2020} & 62.25 & 68.56 & 74.81 & 81.13 & 87.38 & 93.69 \\
\hline
\text{Temperature} \left(^{\circ} C \right) & 2.013 & 19.25 & 2.303 & 19.77 & 2.573 & 20.26 \\
\hline
\text{Days until Hatch} & 189.4 & 38.56 & 177.7 & 37.66 & 168.0 & 36.85 \\
\hline
\end{array}
\][/tex]

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

**References**

Sparks, M. M., Falke, J. A., Quinn, T. P., Adkison, M. D., Schindler, D. E., Bartz, K., Young, D., & Westley, P. A. H. (2019). Influences of spawning timing, water temperature, and climatic warming on early life history phenology in western Alaska Sockeye Salmon. Canadian Journal.