Amarine 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.

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Step-by-step explanation: