Problem 56 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 53-56: Scientists sometimes use the formula $$ f(t)=a \sin (b t+c)+d $$ to simulate temperature variations during the day, with time \(t\) in hours, temperature \(f(t)\) in \({ }^{\circ} \mathrm{C}\), and \(t=0\) corresponding to midnight. Assume that \(f(t)\) is decreasing at midnight. (a) Determine values of \(a, b, c\), and \(d\) that fit the information. (b) Sketch the graph of \(f\) for \(0 \leq t \leq 24\). The high temperature of \(28^{\circ} \mathrm{C}\) occurs at 2 P.M., and the average temperature of \(20^{\circ} \mathrm{C}\) occurs 6 hours later.

Step-by-Step Solution

Answer

Function values: \(a = 8\), \(b = \frac{\pi}{12}\), \(c = -\frac{5\pi}{12}\), and \(d = 20\). Sketch similar to sine wave starting below average, peaking at \(14\), with a 24-hour period.

Step 1: Understand the Problem

We need to determine values for \(a, b, c\), and \(d\) in the sinusoidal function \(f(t) = a \sin(bt + c) + d\) to fit the given temperature variation information. Given the highest temperature of \(28^{\circ} \mathrm{C}\) at \(t = 14\) (2 P.M.), the average temperature of \(20^{\circ} \mathrm{C}\) occurring 6 hours later at \(t = 20\), and knowing \(f(t)\) is decreasing at \(t = 0\) (midnight).

Step 2: Determine the Amplitude and Vertical Shift

The maximum temperature is \(28^{\circ} \mathrm{C}\), and the average or midline temperature is \(20^{\circ} \mathrm{C}\), so the amplitude \(a\) is the difference between the maximum and the average: \(a = 28 - 20 = 8\). The vertical shift \(d\) is the average temperature, which is \(d = 20\). Thus, we have \(f(t) = 8 \sin(bt + c) + 20\).

Step 3: Determine the Period of the Function

The temperature pattern repeats every 24 hours, so the period \(P = 24\) hours. The period of the sine function is given by \(\frac{2\pi}{b} = 24\), solving for \(b\) gives \(b = \frac{\pi}{12}\).

Step 4: Determine the Phase Shift

Given the maximum temperature at \(t = 14\), and knowing the sine function at its maximum at \(\frac{\pi}{2}\), we can determine the phase shift \(c\). Substitute \(f(14) = 8 \sin\left(\frac{\pi}{12} \times 14 + c\right) + 20 = 28\). From \(8 \sin\left(\frac{21\pi}{12} + c\right) = 8\), \(\sin\left(\frac{21\pi}{12} + c\right) = 1\), so \(\frac{21\pi}{12} + c = \frac{\pi}{2}\). Solving for \(c\), we get \(c = \frac{\pi}{2} - \frac{21\pi}{12} = -\frac{5\pi}{12}\).

Step 5: Verify the Function Behavior

To meet all conditions, verify that \(f(t)\) decreases at midnight. Substituting \(t = 0\) into \(f(t)\), we check \(f(0) = 8 \sin(-\frac{5\pi}{12}) + 20\) is decreasing, as the sine function is decreasing initially from a negative phase shift. All conditions including maximum at \(t = 14\) and correct behavior at \(t = 0\) are satisfied.

Step 6: Sketch the Graph

Using the function \(f(t) = 8 \sin\left(\frac{\pi}{12}t - \frac{5\pi}{12}\right) + 20\), plot the graph for \(0 \leq t \leq 24\). The function shows a wave starting below the average temperature at midnight (midnight point), peaking at \(t = 14\) at \(28^{\circ} \mathrm{C}\), troughing around \(8^{\circ} \mathrm{C}\) and returning back to approach ave temperature.

Key Concepts

Temperature Simulation

Temperature simulation often involves modeling how temperatures change over a day. Scientists and mathematicians use sinusoidal functions to mimic these natural patterns because temperature typically fluctuates in waves over the course of a day. This approach helps us understand and predict how temperatures behave, making it useful for various scientific applications, such as weather forecasting and climate studies.

By using the formula \(f(t) = a \sin(bt + c) + d\), we can simulate temperature changes throughout the day, where:

• \(a\) is the amplitude, representing the temperature variation from the average.
• \(b\) determines the stretch or compression of the wave, which affects the period.
• \(c\) is the phase shift, altering the starting point of the wave.
• \(d\) adjusts the average or midline temperature, setting the baseline for the wave.

Through this function, temperature data can be analyzed to provide insights that go beyond mere observation, by allowing us to predict future temperatures based on current trends.

Amplitude

Amplitude in a sinusoidal function represents half of the length between the highest and lowest points in the wave. It is essentially the maximum deviation from the average temperature.

Understanding amplitude helps in determining how extreme the temperature can get from its average.

• In the context of our exercise, the amplitude \(a\) was calculated as \(8\). This means the temperature swings 8 degrees above and below the average temperature of 20 degrees Celsius.
• The amplitude is a crucial value for assessing how wide temperature fluctuations may be during different parts of the day.

With higher amplitudes, the temperature difference between day and night is more significant, while a lower amplitude suggests milder temperature changes.

Period of a Function

The period of a sinusoidal function specifies the time taken for a complete cycle of the wave to occur. In temperature simulations, this time interval is vital to ensure the model reflects daily temperature changes accurately.

The formula for finding the period \(P\) in a sine function is \(\frac{2\pi}{b}\). For the temperature model, we calculate the period as follows:

• Given \(b = \frac{\pi}{12}\), the period \(P = 24\) hours.

This is because the function repeats every 24 hours, imitating the natural cycle of one day. Precise calculation of the period ensures that temperature simulations align with real-world daily fluctuations.

Phase Shift

Phase shift in sinusoidal functions deals with horizontal shifts along the time axis. It's how we adjust where the wave starts relative to time zero, which represents midnight in our scenario. Understanding phase shift is pivotal for aligning models with actual temperature peaks or troughs at specific times.

In our exercise, the phase shift \(c\) is calculated to be \(-\frac{5\pi}{12}\). This shift ensures that:

• The high temperature occurs at 2 PM, aligning the function’s peak with peak temperature times.
• The initial phase ensures that the temperature starts decreasing right at midnight, as given in the problem context.

Having an accurate phase shift aligns predictions with observed data, enhancing the model's reliability as a temperature simulation tool.