- The initial number of fish is \(5\), so the initial value \(a = 5\).
- The population doubles each month, establishing a growth factor of \(b = 2\).
- Using the standard exponential growth formula \(f(x) = a \cdot b^x\), we substitute the identified values to get \(f(x) = 5(2)^x\).
- The final answer is \(\boxed{f(x) = 5(2)^x}\).
Explanation
- Analyzing the Growth Model
The problem describes a population growth scenario where a landscaper introduces an initial population of \(5\) fish into a pond. The population is stated to double every month. We denote \(x\) as the number of months passed since the fish were introduced. Our goal is to determine the function \(f(x)\) that models this population over time. Exponential growth is a common way to model such scenarios where a quantity increases by a constant factor over equal time intervals. The general form of an exponential growth function is \(f(x) = a \cdot b^x\), where \(a\) represents the initial amount and \(b\) represents the growth factor per time unit.
- Determining the Function Coefficients
To define the function, we identify the constants \(a\) and \(b\):
- The initial value \(a\) is the number of fish at the start (at \(x = 0\)). Since the landscaper starts with \(5\) fish, we have \(a = 5\).
- The growth factor \(b\) represents the rate at which the population increases. Since the population doubles each month, the number of fish is multiplied by \(2\) for each increase of \(1\) in \(x\). Thus, the growth factor is \(b = 2\).
Substituting these values into the general exponential form \(f(x) = a \cdot b^x\), we get \(f(x) = 5 \cdot 2^x\).
- Conclusion and Model Verification
We can verify our model by checking values for a few months:
- At \(x = 0\) months, \(f(0) = 5 \cdot 2^0 = 5 \cdot 1 = 5\), which matches the starting population.
- At \(x = 1\) month, \(f(1) = 5 \cdot 2^1 = 5 \cdot 2 = 10\), which is double the initial population.
- At \(x = 2\) months, \(f(2) = 5 \cdot 2^2 = 5 \cdot 4 = 20\), which is double the population at one month.
The function \(f(x) = 5(2)^x\) correctly models the described population growth. Comparing this to the given options, the correct function is \(f(x) = 5(2)^x\).
Examples
Exponential models are essential in fields like biology, finance, and ecology. For instance, if you deposit \(\$100\) into a savings account that doubles in value every year (an unrealistic but illustrative rate!), your money after \(x\) years would be modeled by \(f(x) = 100 \cdot 2^x\). Similarly, scientists use such models to predict the spread of bacteria or the growth of populations, helping them manage resources or prepare for future needs effectively. Understanding these basics helps you visualize how rapid growth functions work in various real-life contexts.