Problem 17 - Calculus: Early Transcendentals

Question

In a certain city the temperature (in \( ^\circ F \)) \( t \) hours after 9 am was modeled by the function $$ T(t) = 50 + 14 \sin \frac{\pi t}{12} $$ Find the average temperature during the period from 9 am to 9 pm.

Step-by-Step Solution

Answer

The average temperature from 9 am to 9 pm is approximately 58.92°F.

Step 1: Convert Time Period to Interval

Since 9 am corresponds to \( t = 0 \) and 9 pm corresponds to \( t = 12 \), the period from 9 am to 9 pm corresponds to the interval \([0, 12]\) in terms of \( t \).

Step 2: Set Up the Integral for Average Temperature

The average temperature over the interval \([0, 12]\) is calculated as \( \frac{1}{12 - 0} \int_{0}^{12} T(t) \, dt \). Substituting the given function, the integral becomes \( \frac{1}{12} \int_{0}^{12} \left( 50 + 14 \sin \frac{\pi t}{12} \right) \, dt \).

Step 3: Break the Integral Down

The integral \( \int_{0}^{12} \left( 50 + 14 \sin \frac{\pi t}{12} \right) \, dt \) can be broken down into two separate integrals: \( \int_{0}^{12} 50 \, dt \) and \( \int_{0}^{12} 14 \sin \frac{\pi t}{12} \, dt \).

Step 4: Evaluate the Constant Integral

The integral \( \int_{0}^{12} 50 \, dt = 50t \Big|_0^{12} \), which evaluates to \( 50(12) - 50(0) = 600 \).

Step 5: Evaluate the Sine Integral

To evaluate \( \int_{0}^{12} 14 \sin \frac{\pi t}{12} \, dt \), use the substitution \( u = \frac{\pi t}{12} \), \( du = \frac{\pi}{12} dt \). The integral becomes \( \int \frac{14 \cdot 12}{\pi} \sin u \, du \). The evaluated integral is \( \frac{-168}{\pi} \cos u \Big|_0^{12} \), which simplifies to \( \frac{-168}{\pi} \left[ \cos \left( \frac{\pi \times 12}{12} \right) - \cos \left( \frac{\pi \times 0}{12} \right) \right] = \frac{-168}{\pi} [(-1) - 1] = \frac{336}{\pi} \).

Step 6: Combine Results for Total Integral Value

Combine the results of the two integrals: \( 600 + \frac{336}{\pi} \).

Step 7: Calculate the Average Temperature

The average temperature is \( \frac{1}{12} \times \left( 600 + \frac{336}{\pi} \right) = 50 + \frac{336}{12\pi} \).

Step 8: Compute Final Value

Compute the numerical approximation for \( \frac{336}{12\pi} \approx 8.92 \), hence the average temperature is approximately \( 50 + 8.92 = 58.92 \).

Key Concepts

Integrals in Calculus

In calculus, integrals are essential tools that help us calculate various values, such as areas under curves and average values over an interval. When we talk about the average temperature calculation for a specific period, we're essentially looking to find the average value of a function over a given range using integrals.

To compute an average of a continuous function like temperature, we use the concept of definite integrals. This involves the following process:

• Identify the interval over which you wish to find the average. For this problem, it was from 9 am to 9 pm, or mathematically, from 0 to 12.
• Next, apply the formula for the average value: \( \frac{1}{b-a} \int_{a}^{b} f(t) \, dt \), where \([a, b]\) is the interval, and \(f(t)\) is the function representing our data.
• Breaking down the integral into simpler parts for easier computation can often simplify the problem.

By calculating each integral separately and then combining them, you achieve the desired average value, giving you insights into the overall behavior of the function over the interval.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate angles to ratios of the sides of right triangles. In physics and engineering, they often model periodic phenomena such as temperature, sound, and light waves due to their wave-like properties.

In this example, temperature is modeled using a sine function, \( T(t) = 50 + 14 \sin \frac{\pi t}{12} \). The sine function is particularly suited here because it varies continuously and smoothly between -1 and 1, which captures the periodic fluctuation in temperatures.

• \(50\) represents the baseline temperature around which fluctuations occur.
• The amplitude, \(14\), shows the degree of fluctuation above and below this baseline.
• The term \(\frac{\pi t}{12}\) alters the period of the sine wave to fit the 12-hour day cycle.

Understanding how trigonometric functions model real-world scenarios is key in mathematical modeling, providing insights into phenomena that cyclically rise and fall.

Mathematical Modeling

Mathematical modeling involves using mathematical structures and relationships to represent real-world systems. By employing mathematical equations and functions, models help predict outcomes and understand complex systems.

In our exercise, the temperature function \(T(t) = 50 + 14 \sin \frac{\pi t}{12}\) is a mathematical model that emulates the daily temperature changes within a city. Mathematical models like this allow us to:

• Predict the future temperature at any given time within the modeled range.
• Understand relationships and potential shifts within the system being studied.
• Make informed decisions based on mathematical predictions.

By integrating concepts like integrals and trigonometric functions into models, we can effectively analyze and interpret real-world data. This synthesis aids in decision-making processes across various domains, such as meteorology and economics.