Abiologist is studying the growth of a particular species of algae. she writes the following equation to show the radius of the algae, f(d), in mm, after d days: f(d) = 7(1.06)d part a: when the biologist concluded her study, the radius of the algae was approximately 13.29 mm. what is a reasonable domain to plot the growth function? (4 points) part b: what does the y-intercept of the graph of the function f(d) represent? (2 points) part c: what is the average rate of change of the function f(d) from d = 4 to d = 11, and what does it represent? (4 points)

F(d) = 7(1.06)^d
part a: when the biologist concluded her study, the radius of the algae was approximately 13.29 mm. what is a reasonable domain to plot the growth function?
solve d for f(d) = 13.29 => 13.29 = 7 (1.06)^d
divide by 7:
(1.06)^d = 13.29 / 7
appply logarithms to both sides => d log(1.06) = log (13.29 / 7)
d = log (13.29/7) / log(1.06) = 11.00
=> the last day, d, was 11.
=> domain = [0, 11]
part b: what does the y-intercept of the graph of the function f(d) represent?
the y-intercept is the value of the function when d = 0, so it represents the length of the algae the day when the study starts.
it is f(0) = 7 (1.09)^0 = 7, which means that the radius of the algae at the moment of starting the study was 7 mm.
part c: what is the average rate of change of the function f(d) from d = 4 to d = 11, and what does it represent?
average rate of change = change in the radius / change in days
average rate of change = [ f(11) - f(4) ] / ( 11 - 4) = [7(1.06)^11 - 7(1.06)^4] / (7)
average rate of change = 4.45 / 7 = 0.64 mm / day
it represents the increase of the radius per day between the days 4 and 11 of the study.

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