Problem 77 - Precalculus: Functions and Graphs

Question

Many animal populations, such as that of rabbits, fluctuate over ten-year cycles. Suppose that the number of rabbits at time \(t\) (in years) is given by $$ N(t)=1000 \cos \frac{\pi}{5} t+4000 $$ (a) Sketch the graph of \(N\) for \(0 \leq t \leq 10\) (b) For what values of \(t\) in part (a) does the rabbit population exceed \(4500 ?\)

Step-by-Step Solution

Answer

The rabbit population exceeds 4500 for \(0 < t < 1.67\) and \(8.33 < t < 10\) years.

Step 1: Understand the Function

The given function for the rabbit population is \(N(t) = 1000 \cos \left(\frac{\pi}{5} t\right) + 4000\). This is a cosine function with amplitude 1000, a vertical shift of 4000, and a period calculated from \(\frac{\pi}{5}\).

Step 2: Determine the Period

The period of the function can be found by determining the value for which \(\frac{\pi}{5} t = 2\pi\). Solving gives \(t = 10\). Thus, the period is 10 years, meaning the function completes one full cycle every 10 years.

Step 3: Plot Key Points

Calculate key points using \(t = 0, 2.5, 5, 7.5, 10\). Substitute these into the function:- \(N(0) = 5000\)- \(N(2.5) = 4000\)- \(N(5) = 3000\)- \(N(7.5) = 4000\)- \(N(10) = 5000\)Graph \(N(t)\) using these points and the known periodic nature.

Step 4: Analyze the Graph

The graph is a cosine wave, starting at 5000 at \(t=0\), descending to 4000 at \(t=2.5\), reaching its minimum at 3000 at \(t=5\), returning to 4000 at \(t=7.5\), and ascending back to 5000 at \(t=10\).

Step 5: Solve for When Population Exceeds 4500

Set \(N(t) > 4500\): \[ 1000 \cos \left(\frac{\pi}{5} t\right) + 4000 > 4500 \]Simplifying, \(\cos \left(\frac{\pi}{5} t\right) > 0.5\). Solve for \(\frac{\pi}{5} t\) yielding: \(-\frac{\pi}{3} < \frac{\pi}{5} t < \frac{\pi}{3}\).

Step 6: Convert Angles to Time Values

Convert the solved inequality to terms of \(t\): Divide by \(\frac{\pi}{5}\):\[-\frac{5}{3} < t < \frac{5}{3} \]Since \(0 \leq t \leq 10\), population exceeds 4500 when \(0 < t < \frac{5}{3}\) and \(10 - \frac{5}{3} < t < 10\).

Step 7: Final Time Interval

Round \(\frac{5}{3}\approx 1.67\), so the rabbit population exceeds 4500 when \(0 < t < 1.67\) and \(8.33 < t < 10\) years.

Key Concepts

Periodic Functions

Understanding periodic functions is crucial when analyzing behaviors that repeat at regular intervals, like the rabbit population in the exercise. In mathematics, trigonometric functions such as sine and cosine are classic examples of periodic functions. These functions repeat their values in predictable cycles.

A function is periodic if there exists a positive number, called a period, such that the function's value repeats every period. In the case of the rabbit population, the cosine function used is periodic because the values of the rabbit count repeat every 10 years, as calculated in the step-by-step solution. The period is denoted by the formula: \[T = \frac{2\pi}{k}\]Here, replaces \(\frac{\pi}{5}\) to give a period of 10 years.

• Periodic functions can model various natural phenomena.
• A full cycle is one complete repetition of the pattern.
• Graphically, they often display as waves.

Working with periodic functions allows us to predict future outcomes based on historical data. This is especially useful for modeling population dynamics, tides, seasonal changes, and more.

Amplitude and Period

Amplitude and period are two important characteristics of trigonometric functions, directly influencing the shape and cycle of the graphs.

• The amplitude is the height from the center line to the peak. This represents half the distance between the maximum and minimum values of the function. In this exercise, the amplitude is 1000, meaning the population count swings 1000 above or below the midline of 4000.



• The period is the length of time it takes for one complete cycle of the function to occur. Calculated as 10 years for our rabbit population, it tells us that the population trend repeats every 10 years.

Understanding these concepts allows us to decompose a trigonometric function into its core properties. This knowledge aids in predicting variations and trends in contexts like animal populations, climate models, and circadian rhythms.

Graphing Trigonometric Functions

Graphing trigonometric functions like cosine and sine helps visualize their properties and how they model real-world scenarios. When you plot these functions, you observe a wave-like pattern, illustrating characteristics such as amplitude, period, and phase shifts.

To graph the function provided, follow these key steps:

• Identify the amplitude and period: Start with understanding the height (amplitude) and the cycle length (period).
• Determine key points: Calculate values at intervals that cover a full period, such as \(t = 0, 2.5, 5, 7.5, 10\).
• Plot the points and sketch the wave: Use key points to outline the wave shape, starting and ending at amplitudes.

Graphing offers a visual representation, making it easier to understand how these mathematical parameters translate to tangible phenomena.

This graph not only provides a depiction of the rabbit population but also aids in identifying when the population exceeds a certain value, as we analyzed in the exercise. Graphs serve as a bridge between abstract mathematical concepts and their practical applications.