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 ⋮ To find the solution set for the equation $P(t) = V(t)$, we need to solve the equation $22.5 + 7.5 \cos\left(\frac{2 \pi(t-28)}{365}\right) = 30.5 + 1.5 \cos\left(\frac{2 \pi(t-90)}{365}\right)$. ‖ ‖ step 2 ⋮ The solution to this equation represents the days of the year when the average high temperatures in Payton's town and Vince's town are equal. ‖ ‖ step 3 ⋮ The asksia-ll calculation list provided the solutions for $t$ which are complex numbers. However, for the context of this problem, we are interested in the real values of $t$ that correspond to the days of the year. ‖ ‖ step 4 ⋮ Since the solutions are given in terms of $n \in \mathbb{Z}$ (the set of all integers), there are multiple days within a year that satisfy the equation, indicating the days when the towns have the same average high temperature. ‖ ***D*** ∻Key Concept∻ ⚹Solution Set Interpretation⚹ ∻Explanation∻ ⚹The solution set for the equation $P(t) = V(t)$ represents the set of all days when the towns have the same average high temperature.⚹◊What is the significance of the solution set for the equation $P(t) = V(t)$ in terms of the average high temperatures for Payton's town and Vince's town?⍭ Generate me a similar question◊

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