Problem 22
Question
The daily temperature high in Minneapolis-St. Paul can be modeled using a formula: $$ T(t)=11.5+7.5 \cos (2 \pi(t-0.6)) $$ where \(T\) is measured in degrees Celsius and \(t\) measures the fraction of the year that has elapsed since January \(1 .\) (a) Find the average annual daily temperature high. (b) Find the time of year (value of \(t\) ) at which this average daily temperature high is actually observed.
Step-by-Step Solution
Verified Answer
(a) 11.5°C; (b) at \(t = 0.35\), early May.
1Step 1: Understand the Function
The given function is \( T(t) = 11.5 + 7.5 \cos(2\pi(t - 0.6)) \). This function calculates the high temperature \( T \) at any time of the year \( t \), where \( t \) is a fraction of the year since January 1. The average temperature is the mean value of \( T(t) \) over the interval from \( t = 0 \) to \( t = 1 \).
2Step 2: Calculate the Average Temperature
To find the average annual temperature, integrate the function \( T(t) \) over the interval from 0 to 1 and divide by the length of the interval. This is computed as follows:\[\text{Average temperature} = \frac{1}{1-0} \int_0^1 T(t) \, dt = \int_0^1 (11.5 + 7.5 \cos(2\pi(t-0.6))) \, dt.\]The integral can be split into:\[\int_0^1 11.5 \, dt + 7.5 \int_0^1 \cos(2\pi(t-0.6)) \, dt.\]
3Step 3: Perform Integration
First, integrate the constant term:\[\int_0^1 11.5 \, dt = 11.5 \times (1 - 0) = 11.5.\]Next, integrate the cosine term. Since the integral of \( \cos \) function over a complete period is zero, i.e.,\[\int_0^1 \cos(2\pi(t-0.6)) \, dt = 0,\]the integral simplifies to:\[7.5 \times 0 = 0.\]
4Step 4: Calculate Total Average
The full average calculation, combining both integrals, is:\[\text{Average temperature} = 11.5 + 0 = 11.5.\]
5Step 5: Analyze When Average Temperature Occurs
The average value is reached when the cosine term itself is zero, i.e., when:\[\cos(2\pi(t-0.6)) = 0.\]This equation is solved by finding:\[2\pi(t-0.6) = \frac{\pi}{2} + n\pi,\]where \(n\) is any integer. Solving this gives\[t - 0.6 = \frac{1}{4} + \frac{n}{2},\]hence:\[t = 0.85, 0.35,\]etc. Checking within the range \(0 \leq t < 1\) gives \(t = 0.35\).
6Step 6: Conclusion
Therefore, the average annual daily temperature high in Minneapolis-St. Paul is 11.5°C, and this average is observed approximately 35% of the way through the year, which is around early May.
Key Concepts
Cosine IntegrationTemperature ModelingAnnual Temperature Cycle
Cosine Integration
When dealing with functions that involve trigonometric components like cosine, integration plays a key role in calculating averages or total values over a certain period. The specific cosine function we are investigating is a part of the formula for temperatures. Understanding the integral of cosine is crucial here.
The integral of the cosine function over a full period, which is from 0 to 1 when considering time in years, becomes zero. This happens because the plus and minus parts of the cosine curve cancel out each other. In our given problem, the integral \( \int_0^1 \cos(2\pi(t-0.6)) \, dt \) evaluates to zero.
This concept illustrates why it’s important to consider boundaries and periods in integration. You’re often looking at an entire cycle, which means anything above the x-axis is canceled out by what is below it, resulting in zero net area. This gives us the conclusion that the cosine part of our temperature formula does not affect the average temperature over the year.
The integral of the cosine function over a full period, which is from 0 to 1 when considering time in years, becomes zero. This happens because the plus and minus parts of the cosine curve cancel out each other. In our given problem, the integral \( \int_0^1 \cos(2\pi(t-0.6)) \, dt \) evaluates to zero.
This concept illustrates why it’s important to consider boundaries and periods in integration. You’re often looking at an entire cycle, which means anything above the x-axis is canceled out by what is below it, resulting in zero net area. This gives us the conclusion that the cosine part of our temperature formula does not affect the average temperature over the year.
Temperature Modeling
Temperature modeling helps us predict or understand temperature trends over time using mathematical formulas. In our exercise, the function \( T(t) = 11.5 + 7.5 \cos(2\pi(t - 0.6)) \) is used to model the daily high temperature in Minneapolis-St. Paul throughout the year.
This function consists of a constant base temperature, 11.5°C, and a fluctuating term, 7.5°C multiplied by a cosine function. The constant part represents the average yearly baseline temperature. While, the cosine term indicates how the temperature varies, due to seasonal changes, adding or subtracting from our base temperature.
By incorporating the cosine function, we can model the typical 'wave' pattern seen in temperature cycles through the year. The amplitude of the cosine term, 7.5, shows the variation from the base temperature, while the phase shift and periodicity are configured to match real-world data closely, such as the yearly temperature cycle.
This function consists of a constant base temperature, 11.5°C, and a fluctuating term, 7.5°C multiplied by a cosine function. The constant part represents the average yearly baseline temperature. While, the cosine term indicates how the temperature varies, due to seasonal changes, adding or subtracting from our base temperature.
By incorporating the cosine function, we can model the typical 'wave' pattern seen in temperature cycles through the year. The amplitude of the cosine term, 7.5, shows the variation from the base temperature, while the phase shift and periodicity are configured to match real-world data closely, such as the yearly temperature cycle.
Annual Temperature Cycle
The annual temperature cycle refers to the regular fluctuations in temperature observed throughout the year. In places like Minneapolis-St. Paul, temperatures rise and fall in a predictable pattern due to the changing seasons.
The function given in the exercise reflects this cyclical nature. It shows how the temperature peaks and troughs align with different times of the year. Here, this is mathematically represented by the term \( \cos(2\pi(t-0.6)) \).
The cycle is defined with a period of one year, meaning it completes one full cycle in this time. The value inside the cosine term, 2𝜋(t-0.6), incorporates both the frequency and a phase shift of 0.6, ensuring the model aligns with the real-world timing of seasons. When we talk about the phase shift of 0.6, we refer to the fact that this curve 'starts' slightly offset to match the actual observed temperature changes, ensuring the model reflects the early rise in spring temperatures. Such models allow better planning and understanding of the climate patterns expected each year.
The function given in the exercise reflects this cyclical nature. It shows how the temperature peaks and troughs align with different times of the year. Here, this is mathematically represented by the term \( \cos(2\pi(t-0.6)) \).
The cycle is defined with a period of one year, meaning it completes one full cycle in this time. The value inside the cosine term, 2𝜋(t-0.6), incorporates both the frequency and a phase shift of 0.6, ensuring the model aligns with the real-world timing of seasons. When we talk about the phase shift of 0.6, we refer to the fact that this curve 'starts' slightly offset to match the actual observed temperature changes, ensuring the model reflects the early rise in spring temperatures. Such models allow better planning and understanding of the climate patterns expected each year.
Other exercises in this chapter
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