Problem 31 - Algebra A Combined Function

Question

Solve. The size of the wolf population at Isle Royale National Park increases according to the formula \(y=y_{0} e^{0.043 t} .\) In this formula, \(t\) is time in years and \(y_{0}\) is the initial population at time 0 . If the size of the current population is 83 wolves, find how many there should be in 5 years. Round to the nearest whole number.

Step-by-Step Solution

Answer

The wolf population in 5 years will be approximately 103 wolves.

Step 1: Identify the given information

We know the current population size is 83 wolves, and this is considered the initial population at time 0, so \(y_0 = 83\). We are asked to find the population after 5 years, which means \(t = 5\).

Step 2: Substitute values into the formula

Substitute \(y_0 = 83\) and \(t = 5\) into the given population growth formula: \[ y = 83 \cdot e^{0.043 \cdot 5} \]

Step 3: Calculate the exponent

First, calculate the product in the exponent: \[ 0.043 \times 5 = 0.215 \]

Step 4: Evaluate the exponential expression

Calculate \(e^{0.215}\). Using a calculator, we find \[ e^{0.215} \approx 1.2407 \]

Step 5: Find the final population

Now substitute \(e^{0.215} \approx 1.2407\) back into the equation for \(y\): \[ y = 83 \times 1.2407 \] Calculate the product to determine the population size.

Step 6: Calculate the final result

Multiply to find the estimated population: \[ y = 83 \times 1.2407 = 102.9781 \] Round the result to the nearest whole number to find the population size in 5 years.

Step 7: Round the answer

Since 102.9781 rounds to 103, there will be 103 wolves in 5 years.

Key Concepts

Population Modeling

Population modeling is a valuable tool used by ecologists to predict how species grow over time. It applies mathematical formulas to model the dynamics of population change. These formulas can help in understanding whether a population will expand, contract, or remain stable.
In the case of the Isle Royale National Park wolf population, the model used is an exponential growth model. Exponential growth occurs when the growth rate of a value is proportional to its current size, leading to faster increases as time goes on. This is typical for populations with abundant resources and little competition or predation.
The formula given for the wolf population is: - \[ y = y_0 e^{0.043t} \]Where:

• \(y\) is the population size at a future time \(t\)
• \(y_0\) is the initial size at time 0
• \(e\) is the base of natural logarithms, approximately equal to 2.718

This model predicts growth over time by introducing a constant growth rate for each time period \(t\). It's important to note that actual populations might not grow exponentially indefinitely due to environmental constraints.

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. In our wolf population model, the constant base is \(e\), an essential part of continuous exponential functions.
In the equation:- \[ y = y_0 e^{0.043t} \]The exponential part \(e^{0.043t}\) governs the growth rate. This function tells us how the population will change over time. In exponential growth, as time \(t\) increases, the population size \(y\) increases rapidly. This behavior is characteristic of natural populations with ideal conditions.

The growth rate is embedded in the exponent, specifically as \(0.043\). This indicates a 4.3% growth rate per year. Calculating this requires careful attention to the exponentiation steps, including understanding how to work with the constant \(e\) using calculators or computational tools.
Exponential functions are pivotal in modeling real-world scenarios, offering a framework to predict natural phenomena like population dynamics.

Rounding Numbers

Rounding numbers is a crucial skill in estimating and making predictions more comprehensible. In real-world applications like population modeling, precise calculations often lead to decimal results.
For the wolf population, the predicted count in 5 years was calculated to be approximately 102.9781. However, since populations are counted in whole units (it isn't practical to have a fraction of a wolf), rounding is necessary. In this case, rounding to the nearest whole number simplifies the prediction to 103 wolves.

To round a number, follow these steps:

• Identify the digit in the tenths place (the first decimal place).
• If this digit is 5 or greater, round the previous number up.
• If it is less than 5, retain the current number.

Rounding ensures that results are both accurate and practical for reporting, especially in studies involving populations, financial figures, or scientific measurements.