A community center started tracking attendance at its annual fundraising event. In the initial year, 2005 $$(t=0)$$, the attendance was $$80$$ people. Five years later $$(t=5)$$, the attendance increased to $$210$$ people. The attendance can be modeled by the function $$A(t)=c+d\ln(t+1),$$ where $$A(t)$$ is the number of attendees and $$t$$ is the number of years since 2005.

(A)

(i) Use the given data to write two equations that can be used to determine the constants $$c$$ and $$d$$.

(ii) Solve for $$c$$ and $$d$$, giving decimal approximations where appropriate.

(B)

(i) Find the average rate of change of attendance from $$t=0$$ to $$t=5$$. Express your answer as a decimal and interpret its meaning in context.

(ii) Use the average rate of change to estimate the attendance at $$t=25$$. The event continued for additional years, and the attendance data is shown below:

(C)

(i) Use a graphing calculator to find a logarithmic regression model for the data. Round coefficients to the nearest thousandth. Explain why a logarithmic model is appropriate.

(ii) Use your model to estimate the attendance at $$t=30$$.

(iii) Explain why the estimate using the average rate of change is greater than the estimate from the logarithmic model.

Section II Part A