Problem 390 - Precalculus

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 \(P(t)=\frac{1000}{1+9 e^{-0.6 t}}\) What is the initial population of fish?

Step-by-Step Solution

Answer

The initial population is 100 fish.

Step 1: Identify the task

We need to determine the initial population of fish in the farm using the given population model equation.

Step 2: Understand initial population

The initial population is the value of the function when time \( t = 0 \) years.

Step 3: Substitute for initial condition

To find the initial population, substitute \( t = 0 \) into the equation: \[P(0) = \frac{1000}{1 + 9 e^{-0.6 \cdot 0}}\]

Step 4: Simplify the expression

Evaluate the expression by simplifying: \[e^{-0.6 \cdot 0} = e^{0} = 1.\] So the equation becomes: \[P(0) = \frac{1000}{1 + 9 \cdot 1}\]

Step 5: Complete the calculation

Complete the calculation by evaluating the denominator: \[P(0) = \frac{1000}{1 + 9} = \frac{1000}{10}\]Therefore, \( P(0) = 100 \).

Key Concepts

Initial Population

The initial population in a population model refers to the number of subjects or entities, in this case, fish, at the starting point of observation. In mathematical terms, it represents the population at time \( t = 0 \). Consider it the baseline from which any growth or decline is measured.

To calculate the initial population using the given formula \[ P(t) = \frac{1000}{1 + 9 e^{-0.6t}}, \]we substitute \( t = 0 \) into the equation.
This simplifies the equation to: \[ P(0) = \frac{1000}{1 + 9 imes e^{0}} = \frac{1000}{10} = 100. \]

Therefore, the initial population of fish in the farm is 100.

This starting point is crucial because it sets the stage for understanding how the population will evolve over time.

• Remember, it provides the basic reference point for analyzing data.
• It's the foundation for extrapolating future trends.

Exponential Function

Exponential functions describe processes of rapid growth or decay. They are essential in modeling real-world situations, including population growth, finance, and even radioactive decay.

In the given population model \( P(t) = \frac{1000}{1 + 9 e^{-0.6t}} \),
the exponential function is evident in the expression \( e^{-0.6t} \).

Here, the base \( e \) denotes the constant approximately equal to 2.718, an important constant in mathematics.
The negative exponent, \( -0.6t \), indicates decay; as time \( t \) increases,
\( e^{-0.6t} \) decreases, leading to an increase in \( P(t) \) due to the form of the equation.

This type of function is common in cases where something grows or diminishes at a rate proportional to its current value.

• Exponential models often represent growth processes.
• Decay models like this one show how systems slow down or reduce over time.

Graphing Calculator

A graphing calculator is a powerful tool for visualizing mathematical equations and functions. It assists in understanding complex concepts such as population models by allowing you to see the changes over time visually.

Using a graphing calculator can help plot the equation:\[ P(t) = \frac{1000}{1 + 9e^{-0.6t}} \]
This approach allows you to observe the behavior of the fish population model over time.

With a graphing calculator, you can:

• Input the equation and generate a graph automatically.
• Explore how the population changes with different values of \( t \).
• Understand the impact of exponential functions visually.

By adjusting parameters and experimenting with different inputs, a graphing calculator enhances your ability to grasp how different elements within an equation affect the overall model.
Additionally, using it repetitively can build your familiarity with mathematical trends and equations.