Problem 30 - Precalculus Mathematics for Calculus

Question

Predator Population Model In a predator/prey model the predator population is modeled by the function $$ y=900 \cos 2 t+8000 $$ where \(t\) is measured in years. (a) What is the maximum population? (b) Find the length of time between successive periods of maximum population.

Step-by-Step Solution

Answer

(a) Maximum population is 8900. (b) Successive maximums occur every \( \pi \) years.

Step 1: Understand the Population Function

The predator population at any given time is given by the function \( y = 900 \cos 2t + 8000 \). Here, \( y \) represents the population and \( t \) is time in years. The function is composed of a cosine term, which is known for its oscillatory behavior, and a constant term that shifts the oscillation vertically.

Step 2: Find the Maximum of the Cosine Function

The cosine function \( \cos 2t \) oscillates between -1 and 1. To find the maximum population, we need the maximum value of the function \( 900 \cos 2t + 8000 \), which happens when \( \cos 2t = 1 \).

Step 3: Calculate Maximum Population

Substitute the maximum value of \( \cos 2t = 1 \) into the population function. This gives us the maximum population as \( 900 \times 1 + 8000 = 900 + 8000 = 8900 \). Therefore, the maximum population is 8900.

Step 4: Determine the Period of the Function

The period of a cosine function \( \cos kt \) is given by \( \frac{2\pi}{k} \). In our function \( \cos 2t \), the coefficient \( k \) is 2. Therefore, the period is \( \frac{2\pi}{2} = \pi \).

Step 5: Convert the Period to Years

Since \( t \) is measured in years and the period of the cosine function is \( \pi \) for \( \cos 2t \), the length of time between successive maximum populations is \( \pi \) years.

Key Concepts

Cosine Function

In this predator-prey model, the predator population is described by the function \( y = 900 \cos 2t + 8000 \). Here, the core component is the cosine function, \( \cos 2t \). The cosine function is a periodic function that oscillates between -1 and 1. It is a fundamental trigonometric function often used to describe cyclical patterns. To understand its role in this model, consider that the cosine term dictates the oscillatory nature of the predator population over time.

Here, the entire function \( 900 \cos 2t + 8000 \) shows us how the predator population changes, with "900" being the amplitude, "2" affecting the frequency of oscillation, and "8000" shifting the function vertically. This setup means the cosine function dynamically affects the population through time.

Maximum Population

To find the maximum population in the given model, we focus on the term \( 900 \cos 2t + 8000 \). The cosine function, \( \cos 2t \), reaches its maximum value of 1 periodically. When it does, it results in the maximum possible value of the population function.

The calculation for the maximum population is straightforward:

• Max of cosine: \( \cos 2t = 1 \)
• Substitute into population function: \( 900 \times 1 + 8000 \)
• Calculate: \( 900 + 8000 = 8900 \)

Thus, the maximum predator population is 8900.

Oscillatory Behavior

Oscillatory behavior is a common characteristic in models that use trigonometric functions, such as our predator population model. The mathematical term "oscillation" refers to a periodic fluctuation or cycle. In this context, the predator population experiences regular ups and downs rather than a steady state, due to the cosine component in the equation.

This oscillation represents the natural rise and fall of predator numbers, driven by interactions like predation and resource availability. The model captures these fluctuations with the help of \( \cos 2t \), inherently demonstrating how populations in natural ecosystems can regularly increase and decrease over time.

Period of a Function

The period of a function describes how long it takes for the function to complete one full cycle. For our cosine function \( \cos 2t \), the period can be calculated using the formula \( \frac{2\pi}{k} \), where \( k \) is the frequency.

• Here, \( k = 2 \).
• Calculate period: \( \frac{2\pi}{2} = \pi \).

The period of \( \cos 2t \) is \( \pi \), which translates to \( \pi \) years in the model. This means it takes \( \pi \) years for the predator population to go from a maximum back to another maximum, completing a full cycle of growth and decline.