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 in this species can be modeled by the function P given by P(t) = abᵗ where P(t) is
the number of plants during year t, and t is the number of years since 2013.
(A) (i) Use the given data to write two equations that can be used to find the values for constants
a and b in the expression for P(t)
(ii) Find the values for a and b as decimal approximations.
(B) (i) Use the given data to find the average rate of change of the number of plants, in plants per
year, from t = 2 to t = 8. Express your answer as a decimal approximation. Show the
computations that lead to your answer.
(ii) Use the average rate of change found in (i) to estimate the number of plants for t = 10 years. Show the work that leads to your answer.
(iii) The average rate of change found in (i) can be used to estimate the number of plants during
year t for t > 10 years. Will these estimates, found using the average rate of change, be less
than or greater than the number of plants predicted by the model P during year t for t > 10 years? Explain your reasoning.
(C) For which t-value t = 6 years or t = 20 years should the ecologist have more confidence in when
using the model P? ive a reason for your answer in the context of the problem.