The function $P(t)$ models the average high temperature, in degrees Celsius, of Paityn's town on the $t^{\text {th }}$ day of the year. The function $V(t)$ models the average high temperature of Vince's town on the $t^{\text {th }}$ day of the year.
$$
\begin{array}{l}
P(t)=22.5+7.5 \cos \left(\frac{2 \pi(t-28)}{365}\right) \
V(t)=30.5+1.5 \cos \left(\frac{2 \pi(t-90)}{365}\right)
\end{array}
$$

Consider the following equation:
$$
22.5+7.5 \cos \left(\frac{2 \pi(t-28)}{365}\right)=30.5+1.5 \cos \left(\frac{2 \pi(t-90)}{365}\right)
$$

What does the solution set for the last equation represent?
Choose 1 answer:
(A) The maximum average high temperature in Paityn's town
(B) The maximum average high temperature in Vince's town
(C) The number of days per year when the towns have the same average high temperature
(D) The set of all days when the towns have the same average high temperature

Question

The function $P(t)$ models the average high temperature, in degrees Celsius, of Paityn's town on the $t^{\text {th }}$ day of the year. The function $V(t)$ models the average high temperature of Vince's town on the $t^{\text {th }}$ day of the year.
$$
\begin{array}{l}
P(t)=22.5+7.5 \cos \left(\frac{2 \pi(t-28)}{365}\right) \
V(t)=30.5+1.5 \cos \left(\frac{2 \pi(t-90)}{365}\right)
\end{array}
$$

Consider the following equation:
$$
22.5+7.5 \cos \left(\frac{2 \pi(t-28)}{365}\right)=30.5+1.5 \cos \left(\frac{2 \pi(t-90)}{365}\right)
$$

What does the solution set for the last equation represent?
Choose 1 answer:
(A) The maximum average high temperature in Paityn's town
(B) The maximum average high temperature in Vince's town
(C) The number of days per year when the towns have the same average high temperature
(D) The set of all days when the towns have the same average high temperature

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ The given equation to solve is $P(t) = V(t)$, which is $22.5 + 7.5 \cos\left(\frac{2 \pi(t-28)}{365}\right) = 30.5 + 1.5 \cos\left(\frac{2 \pi(t-90)}{365}\right)$. This equation will give us the values of $t$ when the average high temperatures in Payton's town and Vince's town are the same. ‖ ‖ step 2 ⋮ According to the asksia-ll calculation list, the solutions to the equation are $t \approx 116.18 + 3.1416n + (0.14524 - 0.27836i)$ and $t \approx 116.18 + 3.1416n + (0.14524 + 0.27836i)$ where $n$ is an element of the set of integers $Z$. This indicates that there are multiple solutions to the equation, corresponding to different days of the year. ‖ ‖ step 3 ⋮ Since the solutions represent multiple days of the year, the solution set of the equation represents all the days when the average high temperatures in Payton's town and Vince's town are the same. ‖ ***D*** ∻Key Concept∻ ⚹Solution Set Interpretation⚹ ∻Explanation∻ ⚹The solution set of the equation $P(t) = V(t)$ represents all the days of the year when the average high temperatures in Payton's town and Vince's town are equal.⚹◊What is the value of the maximum average high temperature in Payton's town according to the given functions $P(t)$?⍭ Generate me a similar question◊

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