Problem 118 - Elementary and Intermediate Algebra

Question

A trap-and-release program run by zoologists found that the ground squirrel population in a wilderness area could be estimated by the logarithmic function \(s(t)=800+600 \log (50 t+1),\) where \(t\) is the number of months after the program started. Find the ground squirrel population 3 years after the program began.

Step-by-Step Solution

Answer

The ground squirrel population is approximately 2753 squirrels after 3 years.

Step 1: Understand the Variable

The variable given in the problem is time, \(t\), measured in months. The problem asks for the population 3 years after the program began. First, we convert years into months since the function \(s(t)\) uses time in months. There are 12 months in a year, so 3 years is \(3 \times 12 = 36\) months.

Step 2: Substitute and Simplify the Argument

Now, substitute \(t = 36\) into the function to find \(s(36)\). The function is \(s(t) = 800 + 600 \log (50t + 1)\). Substitute \(t\) with 36: \(s(36) = 800 + 600 \log (50 \times 36 + 1)\). This simplifies the argument to \(50 \times 36 + 1 = 1800 + 1 = 1801\).

Step 3: Evaluate the Logarithm

Next, evaluate the logarithm \(\log(1801)\). Assuming a common logarithm base 10 is used (as is typical unless specified otherwise), use a calculator to find \(\log(1801)\). This is approximately 3.255.

Step 4: Calculate the Final Result

Use the value of the logarithm to find \(s(36)\): \(s(36) = 800 + 600 \times 3.255\). First, calculate the product: \(600 \times 3.255 = 1953\). Then, add this to the constant 800: \(800 + 1953 = 2753\).

Step 5: Interpret the Result

The ground squirrel population, 3 years after the program began, is approximately 2753 squirrels. This result was found by evaluating the logarithmic function at \(t = 36\) months, corresponding to 3 years.

Key Concepts

Population Estimation

Estimating populations, especially in wildlife conservation, is crucial for understanding ecosystem health and species management. One effective method utilized is mathematical modeling, which presents an analytical approach to predict future trends. Logarithmic functions, like the one given in the exercise, often model real-world phenomena because they accommodate growth that slows as it reaches a certain capacity. In this scenario, the logarithmic function defines how the ground squirrel population changes over time.

• The function provided is: \( s(t) = 800 + 600 \log(50t + 1) \), where \( t \) is time in months.
• The term \( 800 \) can be interpreted as a fixed starting population or baseline.
• The variable \( t \), in months, influences the second part of the function: \( 600 \log(50t + 1) \), modelling how population size changes over time.
• Understanding this function involves converting time (years to months) to make accurate predictions.

By substituting \( t \) with specific values, such as 36 months, you can estimate the ground squirrel population effectively.

Zoology

Zoology is the scientific study of animals, encompassing their behavior, genetics, and ecological roles. In this exercise, zoologists employ the trap-and-release method, a humane and sustainable approach to monitor wildlife populations.
Trap-and-release involves capturing animals from the target species, tagging them for identification, and then releasing them back into their habitat.

• After a period, another set of animals is captured, to see how many tagged ones appear again.
• Based on recapture data, estimates of total population size are made.

Using models and functions, like the logarithmic function provided, helps zoologists predict population changes without extensive capturing or intrusion. It integrates data collection with mathematical calculations, offering a clearer picture while reducing stress on wildlife.

Algebraic Substitution

Algebraic substitution is a fundamental technique in algebra used to solve equations and evaluate functions. It plays a crucial role in finding the value of a function for a given input or variable. The function given for population estimation defines \( t \, \text{in months} \).
Here’s how algebraic substitution works in the context of our example:

• First, convert any given terms to the required unit; convert 3 years to 36 months.
• Substitute the value into the equation: \( s(t) = 800 + 600 \log(50t + 1) \). By replacing \( t \) with 36, we look at \( s(36) \).
• Calculate the inside of the logarithm as \( 50 \times 36 + 1 = 1801 \).
• Solve the logarithm using base 10; approximate \( \log(1801) \approx 3.255 \).
• Use this value to find the population: \( 800 + 600 \times 3.255 \), yielding 2753 squirrels.

Through algebraic substitution, complex functions become manageable, facilitating the calculation of real-world scenarios.