A marine biologist is studying the growth of a particular species of fish she writes the following equation to show the length of the fish f (m) in cm after m months

f (m) = 4 (1.08) m

Part A. when the marine biologist concluded her study the length was approximately 6.86cm. what is a reasonable domain to plot the growth function?

Part b. what does the y intercept of the graph of the function f (m) represent

part c. what is the average rate of change of the function f (m) from m=3 to m=7 and what does it represent?

The given function is:

f (m) = 4 (1.08) ^m

Part A:

solve m when f (m) = 6.86

6.86 = 4 (1.08) ^m

divide both sides by 4:

(1.08) ^m = 6.86 / 4

apply log to both sides:

m log (1.08) = log (6.86 / 4)

m = log (6.86 / 4) / log (1.08)

m = 7 months

Therefore the study concluded after 7 months.

domain = [6.86, 7]

Now solve for f (m) when m = 0

f (m) = 4 (1.08) ^0

f (m) = 4

Therefore the initial length is 4 cm.

domain = [4, 0]

A reasonable domain is between [4, 0] to [6.86, 7].

Part B: The y-intercept is the value of the f (m) when m = 0, so it represents the initial length of the fish at day 0. The y-intercept is:

f (m) = 4 (1.08) ^0

f (m) = 4

Part C:

Average rate of change = change in the radius / change in days

Average rate of change = [ f (7) - f (3) ] / (7 - 3) = [4 (1.08) ^7 - 4 (1.08) ^3] / (7)

Average rate of change = 1.816 / 7

Average rate of change = 0.26 cm / month

This represents the growth per month.