Problem 20 - College Algebra

Question

For the following exercises, use a graphing calculator and this scenario: the population of a fish farm in \(t\) years is modeled by the equation \(P(t)=\frac{1000}{1+9 e^{-0.6 t}} .\) To the nearest whole number, what will the fish population be after 2 years?

Step-by-Step Solution

Answer

The fish population after 2 years will be 270.

Step 1: Understand the Formula

The given equation models the fish population in a fish farm over time. It is given by \( P(t) = \frac{1000}{1 + 9 e^{-0.6t}} \), where \( t \) is the number of years, and \( P(t) \) represents the population at time \( t \). We need to find \( P(2) \), the population at \( t = 2 \) years.

Step 2: Substitute \(t = 2\) into the Formula

Substitute \( t = 2 \) into the equation: \( P(2) = \frac{1000}{1 + 9 e^{-0.6 \times 2}} \). This helps to find the fish population at time \( t = 2 \).

Step 3: Calculate the Exponential Part

Calculate the exponential term \( e^{-0.6 \times 2} \). This simplifies to \( e^{-1.2} \), which needs to be calculated using a calculator to get approximately 0.3012.

Step 4: Perform the Arithmetic Operations

With the calculated value of \( e^{-1.2} \), substitute back in: \( P(2) = \frac{1000}{1 + 9 \times 0.3012} \), which simplifies to \( \frac{1000}{1 + 2.7108} = \frac{1000}{3.7108} \).

Step 5: Final Calculation

Calculate \( \frac{1000}{3.7108} \) using a calculator. This gives a result of approximately 269.5.

Step 6: Round to the Nearest Whole Number

Round 269.5 to the nearest whole number. The nearest whole number is 270.

Key Concepts

Population Growth Models

Population growth models provide valuable insights into how populations change over time. These models help in predicting future populations based on certain assumptions and initial conditions.
They are essential for managing resources, like fish in a fish farm, by providing a systematic way to predict population dynamics.In the fish farm scenario, the population growth model is represented by the equation:

• \( P(t) = \frac{1000}{1+9e^{-0.6t}} \)

This model is a type of logistic growth model, where the growth rate decreases as the population nears its maximum sustainable size, or carrying capacity.
In this equation:

• \( P(t) \) represents the fish population at time \( t \) years,
• \( e \) is the base of the natural logarithm,
• The exponent \(-0.6t\) determines the rate of population change over time.

The population starts small and grows rapidly at first.
As time goes on, the rate of growth slows, illustrating how resources or space in a farm become limited over time. These features make growth models crucial in planning and managing biological resources effectively.

Exponential Functions

Exponential functions are characterized by variables in the exponent. They are written in the form \( f(x) = a \cdot e^{bx} \).
Here, \( e \) is a mathematical constant approximately equal to 2.71828, which represents base exponential processes.In our fish population model, the function involves an exponential element:

• \( e^{-0.6t} \)

This piece specifies how quickly the growth occurs initially and how it tapers off as time progresses.
It shows that as \( t \) (years) increases, the term \( e^{-0.6t} \) decreases, becoming closer to zero.
In other words, the initial rapid growth rate tapers off due to the negative exponent. Studying this behavior helps in understanding natural processes such as population growth, radioactive decay, and compounding interest. These functions are powerful due to their ability to model real-world phenomena that involve growth or decay.

Arithmetic Operations

Arithmetic operations are fundamental mathematical processes, which include addition, subtraction, multiplication, and division.
In our exercise, arithmetic operations are crucial for solving the given formula and projecting the population of the fish farm at \( t = 2 \) years.Here are the arithmetic steps taken:

• Substitute \( t = 2 \): \( P(2) = \frac{1000}{1 + 9e^{-1.2}} \)
• Calculate \( e^{-1.2} \), approximately 0.3012 using a calculator.
• Multiply: \( 9 \times 0.3012 = 2.7108 \).
• Add: \( 1 + 2.7108 = 3.7108 \).
• Divide: \( \frac{1000}{3.7108} \approx 269.5 \).
• Finally, round to the nearest whole number: 270.

Using arithmetic operations properly allows us to handle such equations accurately and reach the desired solution.
These simple operations combined determine the precision of solving population and any other mathematical models effectively.