Question

an ecologist began studying a certain type of plant species in a wetlands area in 2013. in 2015 (t = 2), there were 59 plants. in 2021 (t = 8), there were 118 plants. the number of plants of this species can be modeled by the function p given by p(t) = \int_{0}^{t} a(t) dt, where p(t) is the number of plants during year t, and t is the number of years since 2013. part c in which t - value, t = 6 years or t = 20 years, should the ecologist have more confidence when using the model p? give a reason for your answer in the context of the problem.

Explanation:

The model \( P(t)=\int_{0}^{t} d(t') dt' \) (where \( d(t') \) is the number of plants during year \( t' \) and \( t \) is years since 2013) is based on integrating the growth rate over time. Shorter - term predictions (like \( t = 6 \) years, which is closer to the observed data from \( t = 0 \) to \( t = 8 \)) are generally more reliable because long - term predictions ( \( t = 20 \) years) are more likely to be affected by unmodeled factors (such as environmental changes, new predators, or changes in resource availability) that the integral model may not account for. So the ecologist should have more confidence in the model for \( t = 6 \) years.

Answer:

The ecologist should have more confidence when using the model for \( t = 6 \) years. Because the model is based on observed growth (from \( t = 0 \) to \( t = 8 \)), and long - term (\( t = 20 \)) predictions are more prone to errors from unaccounted environmental or ecological changes, while short - term (\( t = 6 \)) is closer to the observed time frame.