Problem 17 - College Algebra

Question

For the following exercises, use this scenario: The population \(P\) of a koi pond over \(x\) months is modeled by the function \(P(x)=\frac{68}{1+16 e^{-0.22 x}}\). What was the initial population of koi?

Step-by-Step Solution

Answer

The initial population was 4 koi.

Step 1: Understand the function

The given function is \( P(x) = \frac{68}{1 + 16 e^{-0.22 x}} \), which models the population of koi over time. Our task is to find the initial population, which is when \( x = 0 \).

Step 2: Substitute initial value

To find the initial population, substitute \( x = 0 \) into the function. So we calculate \( P(0) = \frac{68}{1 + 16 e^{-0.22 \cdot 0}} \).

Step 3: Simplify the expression

Since \( e^0 = 1 \), the expression becomes \( P(0) = \frac{68}{1 + 16 \cdot 1} = \frac{68}{17} \).

Step 4: Calculate the initial population

Solve \( \frac{68}{17} \), which simplifies to 4. Therefore, the initial population is 4 koi.

Key Concepts

Population Modeling

Population modeling is a fascinating concept that allows us to predict how a group of living organisms, such as koi fish, changes over time. By using mathematical functions, we can simulate and understand the factors affecting growth and decline in populations. In the context of our exercise, the function given is \( P(x) = \frac{68}{1 + 16 e^{-0.22 x}} \). This type of function is often used in exponential growth and logistic modeling.

The logistic function is a common model for population growth as it accounts for the effects of limited resources and environmental pressures, which slow down growth as the population reaches its carrying capacity. This function starts off with rapid growth (exponential growth) when the population is small but gradually slows as it approaches a maximum stable population size, known as the carrying capacity. In this function, the numerator "68" represents this maximum size of the koi population that the pond can support.

• "68" is the carrying capacity—the largest population the environment can sustain.
• The term \( e^{-0.22 x} \) represents the exponential part of the function, influencing growth rate over time \( x \).
• The use of "16" in the denominator affects the initial rapidity of the growth.

By understanding this model, students can learn how to apply mathematical concepts to practical biological scenarios, predicting population behavior based on initial conditions and growth rates.

Initial Population

The initial population of a modeled group is a significant starting point that helps predict how a population will grow over time. To determine this, we substitute the initial time point into the population function, which is often when \( x = 0 \).

In our exercise, we calculate the initial koi population by substituting \( x = 0 \) in the equation: \( P(0) = \frac{68}{1 + 16 e^{-0.22 \cdot 0}} \). This calculation is a simple snapshot of the population size at the beginning of our observation period.

Understanding how to find the initial population is crucial because:

• It provides the baseline for any further growth predictions.
• It's a stepping stone to analyzing the effectiveness of resources and habitat conditions over time.

The solution shows that the initial population is 4 koi by simplifying \( \frac{68}{17} \), showing how math can directly reflect real-world scenarios. It highlights the importance of initial conditions in predicting future outcomes in population modeling.

Koi Pond Population

The population of koi in a pond is an intriguing subject for students fascinated by environmental biology and mathematics. As seen in the exercise, we use a logistic model reflected in \( P(x) = \frac{68}{1 + 16 e^{-0.22 x}} \) to simulate this population.

Koi ponds offer a confined ecosystem where variables such as food supply, space, and water quality heavily influence population growth. The model considers such environmental limits, offering a realistic framework for students to observe:

• How a population starts small and quickly grows under favorable conditions.
• How growth slows and eventually stabilizes as resources become restricted.

Being able to predict how the koi population changes over months allows for better management and understanding of ecological systems. It teaches students how small changes in initial conditions or growth rates may considerably impact long-term predictions and highlights the practical application of mathematical principles in ecological management and conservation.