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

A reasonable domain to plot the growth of the algae's radius function starts at day 0; the y-intercept represents the initial radius of the algae; the average rate of change from day 2 to day 7 reflects the average daily increase in radius in that time span.

Step-by-step explanation:

Part A: Domain of the growth function

To determine a reasonable domain to plot the growth function f(d) for the algae's radius after d days, we can consider the final radius mentioned by the biologist. The algae grew to a radius of approximately 11.79 mm, which suggests that the radius at the beginning of the study, or at day 0, was 11 mm, since f(0) = 11(1.01)0 = 11. A domain starting at d = 0 and extending to the day when the radius reached 11.79 mm would be reasonable. The exact day when the radius was 11.79 mm was not specified, so we cannot determine the upper limit of the domain without further information.

Part B: The y-intercept

The y-intercept of the graph of the function f(d) represents the initial radius of the algae in millimeters when the number of days d is zero.

Part C: Average rate of change

The average rate of change of the function f(d) from d = 2 to d = 7 can be calculated using the formula:

`\[(f(7) - f(2))/(7 - 2)\]`

This represents the average increase in the radius of the algae in millimeters per day over the 5-day period.