- The initial population is \(150\) and it doubles every year, represented by the exponential model \(P(x) = 150 \cdot 2^x\).
- To find when the population reaches \(4,800\), set the function equal to the target value: \(4,800 = 150 \cdot 2^x\).
- Comparing this with the given options, the correct equation is \(4800 = 150(2)^x\).
- Final Answer: \(\boxed{4800 = 150(2)^x}\)
Explanation
- Analyzing the Population Growth Model
The problem describes the population growth of rabbits on an island. The initial population on January 1 is \(P_0 = 150\). We are told that the population doubles every year. Exponential growth follows the model \(P(x) = P_0 \cdot r^x\), where \(P(x)\) is the population after \(x\) years, \(P_0\) is the initial population, and \(r\) is the growth factor per time period. Since the population doubles every year, the growth factor \(r = 2\). Substituting the given values into the model gives \(P(x) = 150 \cdot 2^x\). We want to find the number of years \(x\) such that the population reaches \(4,800\). Setting \(P(x) = 4,800\), we obtain the equation \(4,800 = 150 \cdot 2^x\).
- Evaluating the Given Options
Comparing our derived equation \(4800 = 150(2)^x\) with the given choices:
- \(4800 = 150(2)^x\) matches our result.
- \(4800 = 2(150)^x\) implies the growth factor is 150, which is incorrect.
- \(4800 = 2x + 150\) describes a linear relationship, which does not model exponential growth (doubling).
- \(4800 = 2^x + 150\) suggests the doubling happens relative to an addition, not a multiplication.
Therefore, the correct equation to determine the number of years \(x\) is \(4800 = 150(2)^x\).
Examples
Exponential growth models are used in many fields to track how quantities increase over time. For example, in finance, interest compounding is modeled using exponential growth, helping people understand how their savings grow over many years. By understanding these formulas, you can predict future values for investments, population sizes, or even the spread of information on social media. Accurate models allow for better planning and decision-making in real-world scenarios.