Problem 108 - Precalculus

Question

For Exercises 107 and \(108,\) refer to the following: With the advent of summer come fireflies. They are intriguing because they emit a flashing luminescence that beckons their mate to them. It is known that the speed and intensity of the flashing are related to the temperature- the higher the temperature, the quicker and more intense the flashing becomes. If you ever watch a single firefly, you will see that the intensity of the flashing is periodic with time. The intensity of light emitted is measured in candelas per square meter (of firefly). To give an idea of this unit of measure, the intensity of a picture on a typical TV screen is about 450 candelas per square meter. The measurement for the intensity of the light emitted by a typical firefly at its brightest moment is about 50 candelas per square meter. Assume that a typical cycle of this flashing is 4 seconds and that the intensity is essentially zero candelas at the beginning and ending of a cycle. Graph the equation from Exercise 107 for a period of 30 seconds.

Step-by-Step Solution

Answer

Graph a sinusoidal function with a period of 4 seconds for 30 seconds.

Step 1: Understand the Problem

We are given a problem related to the intensity of light emitted by a firefly, which varies periodically over a cycle of time. The intensity is measured in candelas per square meter, with a maximum intensity of 50 candelas per square meter during the brightest moment of the cycle, which lasts for 4 seconds per cycle. We need to graph this intensity over a period of 30 seconds.

Step 2: Define the Intensity Function

Since the intensity of light is periodic and peaks at 50 candelas per square meter, we can assume it follows a sinusoidal pattern. The general form for the intensity function could be modeled as \( I(t) = 25 \sin\left(\frac{\pi}{2} t \right) + 25 \), where \(t\) is time in seconds. This matches the periodic nature with a 4-second cycle and ensures that intensity goes from 0 to 50 candelas.

Step 3: Determine the Pattern for 30 Seconds

One cycle occurs every 4 seconds, so over 30 seconds, there will be multiple cycles. Specifically, there will be \(\frac{30}{4} = 7.5\) cycles. We will graph the function up to \(t = 30\).

Step 4: Plot the Intensity Function

Using the defined function \( I(t) = 25 \sin\left(\frac{\pi}{2} t \right) + 25 \), plot the graph over the time period from 0 to 30 seconds. For each 4-second interval, the intensity should start at 0 candelas, reach 50 candelas, and return to 0 candelas, repeating this wave-like pattern for the entire duration.

Step 5: Verify the Graph Characteristics

Ensure that the graph shows the correct sinusoidal pattern, with peaks occurring every 2 seconds (midway through each 4-second cycle) and returning to 0 at the start and end of each 4-second cycle. The graph should resemble a wave with 7 and a half complete oscillations over the 30-second period.

Key Concepts

Sinusoidal Functions

Sinusoidal functions are a special type of periodic functions that repetitively oscillate between a maximum and a minimum value. These functions are defined by the sine and cosine equations in trigonometry, which create smooth and continuous wave-like patterns. The characteristic shape of a sinusoidal function is derived from these repetitive cycles.
\( I(t) = A \sin(Bt + C) + D \) is one common form where:

• \(A\) represents the amplitude (peak height/half the peak difference).
• \(B\) defines the frequency, impacting how quickly cycles repeat.
• \(C\) is the phase shift, which shifts the wave left or right.
• \(D\) vertically shifts the wave up or down.

In our problem with the firefly, the function used is \( I(t) = 25 \sin\left(\frac{\pi}{2} t \right) + 25 \).
This function is customized to fit the unique behavior of the firefly's light emissions.
The sinusoidal properties make it possible for us to predict the repeating pattern of light intensity emitted by the firefly.

Graphing

Graphing a sinusoidal function involves plotting the values of the function over a set period on a coordinate plane. This visual representation helps us better understand how the function behaves over time. When working with sinusoidal functions, certain steps can be followed to ensure accurate graphing.
First, determine the cycle and period. In our exercise, the flashing of the firefly follows a 4-second cycle. Each cycle can be segmented into its individual rise and fall, reaching the maximum intensity at 2 seconds and returning to 0 at 4 seconds.

Second, decide on the timing for the graph. Here, we plotted our equation \( I(t) = 25 \sin\left(\frac{\pi}{2} t \right) + 25 \) over a period of 30 seconds. This means that the graph should complete 7.5 cycles within this duration.
When plotting, marks are placed every 4 seconds along the time axis, and intensity is marked from 0 to 50 on the vertical axis. By connecting these points smoothly, we visualize the oscillations over time.
Graphing like this not only shows us predicted values but also helps us notice trends, peaks, and troughs.

Trigonometric Functions

Trigonometric functions include sine, cosine, and tangent, all stemming from the study of triangles and angles. These functions are incredibly useful in representing periodic phenomena due to their wave-like nature.
In our firefly problem, the sine function was specifically chosen to mirror the periodic intensity pattern. The sine function is fundamentally based on the idea of repeated motion, making it perfect for modeling things like sound waves, tides, or light as in our firefly.

The reason trigonometric functions are needed is because they naturally cycle between -1 and 1, capturing the maxima and minima qualities we see in periodic phenomena. In the context of the firefly's flashing, the sine function controls how intensity rises to a peak and falls back to zero within a cycle.
By adjusting parameters like amplitude and frequency, trigonometric functions become robust tools in creating accurate models for any scenario involving cycles and regular repetitive patterns.
This modeling capability is what makes them so essential in science and engineering applications when describing the world around us.