A recent study has collected data on the daily temperature variations in a region over different parts of the day. The trend shows that the temperature increases steadily during the morning, remains relatively stable during the afternoon, and decreases sharply in the evening.
Based on the description below, answer the following:
(a) Determine appropriate time intervals for each segment of the day with distinct temperature behavior.
(b) Propose a piecewise function \(T(t)\) that models the temperature \(T\) in degrees Celsius as a function of time \(t\) (in hours), where each function segment corresponds to the intervals identified in part (a). Assume a linear model in each segment.
(c) Discuss how you would adjust the model to ensure continuity at the transition times between segments.
Question
Identify and justify time intervals for the morning increase, afternoon stability, and evening decrease in temperature based on the provided description.
Hint
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Hint
Think of typical hours for morning, afternoon, and evening.
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Complete Answer
One plausible set of intervals is: Morning from 6 AM to 12 PM (6 \(\leq t < 12\)), Afternoon from 12 PM to 6 PM (12 \(\leq t < 18\)), and Evening from 6 PM to 10 PM (18 \(\leq t \leq 22\)).
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Question
Construct a piecewise function \(T(t)\) that models the temperature using linear expressions for each time interval identified in part (a). Use the general forms \(T(t)=m_1t+c_1\) for the morning, \(T(t)=m_2t+c_2\) for the afternoon, and \(T(t)=m_3t+c_3\) for the evening. Explain what the parameters \(m_1, m_2, m_3, c_1, c_2, c_3\) represent in the context of temperature modeling.
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Hint
Consider the slope and y-intercept interpretation for linear functions in each interval within the context of temperature change over time.
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A valid model is: \(T(t)=\begin{cases} m_1t+c_1, & 6\leq t<12 \\ m_2t+c_2, & 12\leq t<18 \\ m_3t+c_3, & 18\leq t\leq22 \end{cases}\). Here, \(m_1\) represents the rate of temperature increase in the morning, \(m_2\) the rate of temperature change in the afternoon (ideally close to zero for stability), and \(m_3\) the rate of temperature decrease in the evening. \(c_1, c_2, c_3\) are the y-intercepts, which, when considering the domain starting from t=6 for the first segment, are less directly interpretable without further constraints but help define the temperature level within each interval.
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Question
Discuss how you might adjust the function model to ensure continuity at the boundaries between the time intervals. Provide the equations that must be satisfied to ensure continuity at \(t=12\) and \(t=18\). Explain what these conditions mean for the temperature model.
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Hint
For continuity at a point, the function values from both sides must be equal at that point. Apply this to the boundaries of your intervals.
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To ensure continuity at \(t=12\), we need to ensure that the morning temperature function and the afternoon temperature function meet at \(t=12\). This gives the condition: \(m_1(12)+c_1 = m_2(12)+c_2\). Similarly, for continuity at \(t=18\), the afternoon and evening functions must meet, giving: \(m_2(18)+c_2 = m_3(18)+c_3\). These conditions mean that there are no abrupt jumps in temperature at the transition times between morning and afternoon, and afternoon and evening, resulting in a smooth, continuous temperature profile throughout the modeled day.
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