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) = 9(1.04)d

Part A: When the biologist concluded her study, the radius of the algae was approximately 12.81 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 = 3 to d = 9, and what does it represent?

Answer 1

A. 0 ≤ d ≤ 9

B. the initial radius of the algae

C. about 0.448; the average growth rate of the radius in mm/day

Explanation:

You want to know the reasonable domain, the meaning of the y-intercept, and the average rate of change of the function f(d) = 9(1.04^d), representing the radius in mm after d days of a growth of algae.

A. Domain

The study was concluded at the point in time when the radius of the growth was 12.81 mm. That corresponds to d = 9. The active period of the study, hence the reasonable domain of the function, was ...

0 ≤ d ≤ 9 . . . . reasonable domain of f(d)

B. Y-intercept

The y-intercept is f(0) = 9. The function definition tells you this is the radius of the growth after 0 days.

The y-intercept (9 mm) is the initial radius of the algae growth.

C. Average rate of change

The average rate of change from d=3 to d=9 is given by ...

(f(9) -f(3))/(9 -3) ≈ (12.810 -10.124)/6 ≈ 0.448 . . . . mm/day

The average rate of change of f(d) was about 0.448 mm/day.

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Additional comment

The y-intercept is where the function graph crosses the y-axis. The value of x there is x=0, so the function value f(0) gives you the y-intercept.

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