Problem 18
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
Verified Answer
The initial population of fish is 100.
1Step 1: Understanding the Problem
We need to find the initial population of fish in the farm. The initial population refers to the population at time \( t = 0 \).
2Step 2: Substitute Initial Condition
To find the initial population, substitute \( t = 0 \) into the given population function \( P(t) = \frac{1000}{1 + 9e^{-0.6t}} \).
3Step 3: Simplify the Exponential Term
Calculate the value of the exponential term when \( t = 0 \): \( e^{-0.6 \times 0} = e^0 = 1 \).
4Step 4: Evaluate the Population Formula
Substitute the simplified value back into the formula: \( P(0) = \frac{1000}{1 + 9 \times 1} = \frac{1000}{1 + 9} = \frac{1000}{10} \).
5Step 5: Final Calculation
Perform the division: \( \frac{1000}{10} = 100 \). Hence, the initial population is 100 fish.
Key Concepts
Initial PopulationExponential FunctionGraphing Calculator Usage
Initial Population
In mathematical models, the initial population refers to the number of individuals present at the start of observation, when time is zero. It's crucial when exploring population dynamics, as it acts as a baseline measurement for future calculations.
To determine this, you simply substitute time, denoted as \( t \), with zero in the population equation. This substitution simplifies any exponential growth model calculations.
For instance, in the given scenario, we substitute zero into the function \( P(t) = \frac{1000}{1 + 9e^{-0.6t}} \). At \( t = 0 \), the expression \( e^{-0.6 \times 0} \) simplifies to 1 because anything raised to the power of zero is 1.
After substituting this back into the function, it becomes \( P(0) = \frac{1000}{1 + 9 \times 1} = \frac{1000}{10} = 100 \).
To determine this, you simply substitute time, denoted as \( t \), with zero in the population equation. This substitution simplifies any exponential growth model calculations.
For instance, in the given scenario, we substitute zero into the function \( P(t) = \frac{1000}{1 + 9e^{-0.6t}} \). At \( t = 0 \), the expression \( e^{-0.6 \times 0} \) simplifies to 1 because anything raised to the power of zero is 1.
After substituting this back into the function, it becomes \( P(0) = \frac{1000}{1 + 9 \times 1} = \frac{1000}{10} = 100 \).
- This tells us that at the start (\( t = 0 \)) there are 100 fish.
Exponential Function
Exponential functions are fascinating mathematical expressions characterized by continuous growth or decay. They're widely used in biology, economics, and many other fields to model changes over time.
They have the general form \( P(t) = a \cdot e^{rt} \), where \( a \) is the initial quantity, \( e \) is the base of natural logarithms, and \( r \) is the rate of growth or decay.
In our fish farm scenario, the exponential expression \( 9e^{-0.6t} \) depicts how quickly the fish population changes over time, with \( -0.6 \) indicating a decay when factoring in the environment limiting factors.
The negative exponent implies that as time progresses, the function results in a decreasing rate of growth, which is typical in situations where resources are limited or there is competition.
They have the general form \( P(t) = a \cdot e^{rt} \), where \( a \) is the initial quantity, \( e \) is the base of natural logarithms, and \( r \) is the rate of growth or decay.
In our fish farm scenario, the exponential expression \( 9e^{-0.6t} \) depicts how quickly the fish population changes over time, with \( -0.6 \) indicating a decay when factoring in the environment limiting factors.
The negative exponent implies that as time progresses, the function results in a decreasing rate of growth, which is typical in situations where resources are limited or there is competition.
- An exponential function like this is foundational in modeling realistic population dynamics.
Graphing Calculator Usage
A graphing calculator is an invaluable tool in visualizing and solving complex functions like exponential equations. These devices or software allow you to plot functions, providing a visual interpretation of mathematical models.
Using a graphing calculator, you can input equations and instantly see how variables interact over time. For exponential growth problems, you'll see curves that sharply increase or level off.
This visual aid helps you understand concepts that are otherwise abstract, such as limits in growth due to environmental factors, seen in the flattening curve of our fish population model.
Using a graphing calculator, you can input equations and instantly see how variables interact over time. For exponential growth problems, you'll see curves that sharply increase or level off.
This visual aid helps you understand concepts that are otherwise abstract, such as limits in growth due to environmental factors, seen in the flattening curve of our fish population model.
- Enter the function \( P(t)=\frac{1000}{1+9e^{-0.6t}} \) into the calculator.
- Adjust the viewing window to capture both initial and long-term behaviors of the population.
- Analyze how changes in time \( t \) affect the population size \( P(t) \).
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