At the start of a research study, a colony of penguins had a population of 20,000. One year later, it had a population of 21,200.
Assuming the population of the colony has grown exponentially, which expression best models the population? Let t represent the time in years from the start of the research study.
Assuming the colony continues to grow at the same rate, what will the population of the colony be 4 years after the start of the research study?
Round your answer to the nearest whole number.
penguins
The population of the colony can be modeled by the equation P(t) = P_0 * e^(rt), where P(t) is the population at time t, P_0 is the initial population, e is the base of the natural logarithm (approximately 2.71828), and r is the growth rate.
To find the growth rate, we can use the formula r = ln(P(t)/P_0) / t, where ln denotes the natural logarithm. Plugging in the values P(t) = 21,200, P_0 = 20,000, and t = 1, we get:
r = ln(21,200/20,000) / 1 ≈ 0.05657
Therefore, the equation that best models the population is P(t) = 20,000 * e^(0.05657t).
To find the population 4 years after the start of the research study (t = 4), we can plug in t = 4 into the equation:
P(4) = 20,000 * e^(0.05657*4) ≈ 20,000 * e^(0.22628) ≈ 20,000 * 1.25309 ≈ 25,061.8
Rounding to the nearest whole number, the population of the colony 4 years after the start of the research study would be approximately 25,062.