Question 1 (essay worth 45 points) (04.03 mc) 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? part b: what does the y-intercept of the graph of the function f(d) represent? 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?

Function: radius = f(d) = 9 * [1.04] ^ d

A: domain

It is the possible values of d, which are from 0 (just before the study starts) to  the value d that makes f(d) = 12.81

=> 12.81 = 9 * [1.04]^d => [1.04]^d  = 12.81 / 9 = 1.4233

Take ln to both sides to find d:

d ln[1.04] = ln [1.4233] => d = ln [1.4233] / ln [1.04] = 9.00

So, the study lasts until the day 9, and the domain is the interval [0, 9]

B) The y - intercept is the value of the function when d = 0, so it represents the radius of the algae at start.

It is f(0) = 9 * [1.04]^0 = 9. Which means that the initial radius of the algae is 9 mm.

C) find the values of the function when d = 3 and when d = 9.

d= 3 => f(3) = 9 *[1.04]^3 = 10.12 mm

d = 9 => f(9) = 12.81 mm     

rate of change = [change in radius] / [number of days] = 

rate of change = [12.81mm - 10.12mm] / [9 days - 3 day] = 2.69 mm / 6 days

rate of change = 0.448 mm/days.

It represents the average change of radius per day.