Problem 117 - Elementary and Intermediate Algebra

Question

A farmer stocked a lake on her property with 75 sunfish. She was told that with the proper oversight, the sunfish population could be approximated by the logarithmic function \(f(t)=75+45 \log (t+1),\) where \(t\) is the number of years since the lake was stocked. Find the number of sunfish she can expect in the lake in \(2 \frac{1}{2}\) years.

Step-by-Step Solution

Answer

The farmer can expect approximately 99 sunfish in the lake after 2.5 years.

Step 1: Understand the problem

The problem asks us to find the expected number of sunfish in the lake after 2.5 years, given a logarithmic model that describes the population growth over time.

Step 2: Identify the given function

We are given the function that models the sunfish population: \(f(t) = 75 + 45 \log(t+1)\), where \(t\) is the number of years.

Step 3: Substitute the given time into the function

Given \(t = 2.5\) years, substitute it into the function to find the number of sunfish: \(f(2.5) = 75 + 45 \log(2.5+1)\).

Step 4: Simplify the expression inside the logarithm

Calculate \(2.5 + 1\) which is equal to 3.5. So, we now have: \(f(2.5) = 75 + 45 \log(3.5)\).

Step 5: Calculate the logarithm

Use a calculator to find the value of \(\log(3.5)\). The common logarithm (base 10) of 3.5 is approximately 0.5441.

Step 6: Compute the scaled logarithm result

Multiply the logarithm by 45: \(45 \times 0.5441 = 24.4845\).

Step 7: Add the initial population to the scaled result

Add the initial number of sunfish to the result from Step 6: \(75 + 24.4845 = 99.4845\).

Step 8: Round the answer to a whole number

Since the number of sunfish must be a whole number, round 99.4845 to 99.

Key Concepts

Population Growth Models

Population growth models are mathematical equations used to describe how a population increases or decreases over time. These models help us understand and predict future population sizes in different environments. For instance, the farmer's sunfish population is modeled with the logarithmic function.

In this case, we use the function:

• \[f(t) = 75 + 45 \log(t+1)\]
• Here, \(f(t)\) represents the population of sunfish after \(t\) years
• 75 is the initial number of sunfish present when the lake was first stocked
• 45 is the growth factor connected to the logarithmic part that depicts how fast the population could grow

Logarithmic growth models are useful because they show how populations can grow rapidly at first and then slow down as they approach a certain limit. This pattern is often seen in real-life scenarios, such as a lake that starts with a small fish population but needs time and resources to sustain larger numbers.

Logarithmic Equations

Logarithmic equations involve the logarithm function, a mathematical tool used to solve equations that involve exponential growth or decay. In the given function,
\[f(t) = 75 + 45 \log(t+1)\]we see a logarithmic term with base 10 implicitly stated. Logarithms are the inverse functions of exponentials, and they are incredibly useful for handling large numbers in a compressed form.

To solve the farmer's problem, we substitute the given value of \(t = 2.5\) into the function:

• Calculate inside the logarithm: \(t+1 = 3.5\)
• Find the logarithm: \(\log(3.5)\) which is approximately 0.5441
• Multiply the logarithmic result by 45: \(45 \times 0.5441 = 24.4845\)

Understanding logarithms is crucial as they turn multiplication processes into addition, making complex calculations more manageable. This can especially be helpful when predicting natural phenomena like population growth.

Rounding Numbers

Rounding numbers is about simplifying them, making them easier to comprehend and use in further calculations. This is especially useful when the result needs to be a whole number or is given in a context where precision isn't crucial to a larger extent.

In our population model, after computing \[75 + 24.4845 = 99.4845\]we notice the result is not a whole number. Since we can't have a fraction of a sunfish, we need to round 99.4845 to 99.

There are several rounding conventions, but a common one is rounding to the nearest whole number. Here's how it works:

• If the number after the decimal is 5 or more, you round up.
• If it is less than 5, you round down.

Thus, applying rounding provides us a clean, whole number answer that is both easier to communicate and more practical in real-world applications.