Question

A town with a population of 200 people grows to a population of 400 in six years. Assume the population P grows exponentially according to the equation: a) P = 200 * e^(0.115t) b) P = 200 * e^(0.069t) c) P = 200 * e^(0.046t) d) P = 200 * e^(0.023t)

Answer 1

The correct equation that represents the population growth in the town is P = 200 * e^(0.069t). This equation represents exponential growth with a doubling time of approximately 10 years.

Step-by-step explanation:

The equation that represents the population growth in the town is given by:

P = 200 * e^(0.069t)

Option b) is the correct answer.

This equation represents exponential growth, where P is the population at time t. The constant 0.069 represents the growth rate, which corresponds to a doubling time of approximately 10 years (by the rule of 70). Thus, the town's population will double every 10 years.

Answer 2

To solve the student's problem, we must find which exponential growth equation correctly models the town's population doubling from 200 to 400 in 6 years. By testing each equation, we can determine that option b) P = 200 * e^(0.069t) is the correct one when we plug in t=6 and get approximately 400.

Step-by-step explanation:

The student's question pertains to exponential population growth, and they need to understand how to calculate the population at a given time with a specific growth rate using an exponential growth formula. The provided equations represent different rates of exponential growth, denoted by e raised to the power of a growth rate multiplied by time (t). To determine which of the provided equations accurately describes the growth from 200 to 400 people in 6 years, we use each equation and plug in the values for time to see if the output equals 400.

For example, if we take option a), we replace t with 6 to get P = 200 * e^(0.115*6). If this yields 400, then the equation correctly represents the growth in question. However, this calculation results in a number that exceeds 400, indicating that the growth rate in option a) is too high. Similarly, we can test each option and find that the correct growth rate is option b), P = 200 * e^(0.069*6), which results in a population of approximately 400 in 6 years.

This exercise emphasizes the significance of understanding exponential functions and growth rates in analyzing population dynamics, reflecting trends observed in real-world scenarios where growth can either be unrestricted or eventually level off due to resource constraints.