Question

A biologist 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) = 11(1.01)d

Part A: When the biologist concluded her study, the radius of the algae was approximately 11.79 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 = 2 to d = 7, and what does it represent? (4 points)

Answer 1

The reasonable domain to plot the growth function is any positive real numbers, the y-intercept represents the initial radius of the algae, and the average rate of change represents the average rate of increase in the radius of the algae per day.

Step-by-step explanation:

Part A: The domain of the growth function represents the set of all possible input values, which in this case is the number of days. Since the biologist concluded her study when the radius of the algae was approximately 11.79 mm, a reasonable domain to plot the growth function would be any positive real numbers, as it is possible for the radius to continue increasing beyond the observed value.

Part B: The y-intercept of the graph of the function f(d) represents the value of the function when the input variable, in this case, the number of days, is zero. In the given equation f(d) = 11(1.01)^d, plugging in d=0 gives f(0) = 11(1.01)^0 = 11. Therefore, the y-intercept represents the initial radius of the algae.

Part C: To find the average rate of change of the function f(d) from d=2 to d=7, we need to calculate the difference in function values and divide by the difference in days. Substituting the values in the equation, f(2) = 11(1.01)^2 and f(7) = 11(1.01)^7, the average rate of change can be calculated as (f(7) - f(2))/(7-2). This value represents the average rate of increase in the radius of the algae per day from day 2 to day 7.