Problem 19 - 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 tenth, what is the doubling time for the fish population?

Step-by-Step Solution

Answer

The doubling time is approximately 1.4 years.

Step 1: Understand the Problem

We need to find the time it takes for the fish population to double. The initial population is found at time \( t = 0 \). The population we will aim to achieve is double this initial value.

Step 2: Determine Initial Population

Calculate the initial population, \( P(0) \), by substituting \( t = 0 \) into the formula: \[ P(0) = \frac{1000}{1 + 9e^{-0.6 \times 0}} = \frac{1000}{1 + 9 \times 1} = \frac{1000}{10} = 100. \]This means the initial population is 100 fish.

Step 3: Set Up Equation for Double the Population

We want the population to double, so we set \( P(t) = 200 \), because \( 2 \times 100 = 200 \). Plug \( P(t) = 200 \) into the equation:\[ 200 = \frac{1000}{1 + 9e^{-0.6t}}. \]

Step 4: Solve for \(t\)

First, solve the equation for \( t \) by cross multiplying to get:\[ 200(1 + 9e^{-0.6t}) = 1000. \]\[ 200 + 1800e^{-0.6t} = 1000. \]Subtract 200 from both sides:\[ 1800e^{-0.6t} = 800. \]Divide both sides by 1800:\[ e^{-0.6t} = \frac{800}{1800}. \]Simplify:\[ e^{-0.6t} = \frac{4}{9}. \]Take the natural logarithm of both sides:\[ -0.6t = \ln\left(\frac{4}{9}\right). \]Solve for \( t \):\[ t = \frac{\ln\left(\frac{4}{9}\right)}{-0.6}. \]

Step 5: Calculate \(t\) Using a Calculator

Using a graphing calculator, compute the natural logarithm and the division:\[ t \approx \frac{-0.8109}{-0.6} \approx 1.3515. \]Round to the nearest tenth: \( t \approx 1.4 \).

Key Concepts

Doubling Time

When you hear the term "doubling time," it refers to the amount of time it takes for a quantity to increase by 100%. In our fish farm scenario, we begin with a specific fish population and aim to find when it will double.
To start, you need to calculate the initial population using the function provided. Once the initial population is determined, the next step is to double that number.
This value will be our target population. You then use the same population equation, but now substitute this new target population to solve for the time, denoted as \( t \).
In general:

• Initial population: From the function at \( t = 0 \).
• Target population: Twice the initial population.
• Equation setup: Substitute the target into the original equation and solve for \( t \).

This step-by-step approach helps you understand how the doubling time directly connects with exponential growth in populations.

Population Modeling

Population modeling is a mathematical approach used to represent how a population changes over time. For phenomena like our fish farm, an exponential model is often applied.
Our given equation models the population by using parameters and an exponential expression. This helps to describe the growth influenced by time, \( t \).
Key aspects to understand in population modeling include:

• **Parameters in the function**: Such as the numbers 1000 and 9, which influence the behavior of the population curve.
• **Exponential factor**: The term \( e^{-0.6t} \) describes the exponential nature of the growth.
• **Time as a variable**: As time progresses, the population changes according to the model.

By understanding these components, you can assess how a population evolves and how long certain milestones, such as doubling the initial size, will take.

Graphing Calculator Usage

Using a graphing calculator can immensely simplify solving complex equations like those in population modeling.
When calculating the doubling time, it is critical to compute precise math operations, such as logarithms and exponents.
Here's how you effectively use a graphing calculator:

• Enter complex expressions accurately: For instance, typing \( \ln\left(\frac{4}{9}\right) \) in your calculator precisely ensures you get the correct logarithm value.
• Perform division and other operations: It helps to compute operations like dividing the logarithm by \(-0.6\) to find \( t \).
• Use functions and built-in features: Most calculators have a log function or even a solve feature you can exploit for faster results.

Utilizing a graphing calculator efficiently will not only give you the right answer but also hone your skills in calculations that are frequent in math-related fields.