Question

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) = 5(1.07)m

Part A: When the marine biologist concluded her study, the length of the fish was approximately 9.19 cm. 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(m) represent? (2 points)

Part C: What is the average rate of change of the function f(m) from m = 1 to m = 9, and what does it represent? (4 points)

Answer 1

Given : A marine biologist is studying the growth of a particular species of fish

$f(m) = 5 (1.07)^m$

: When the marine biologist concluded her study,  the length of the fish was approximately 9.19 cm

To Find : reasonable domain to plot the growth function

What does the y-intercept of the graph of the function f(m) represent?

average rate of change of the function f(m) from m = 1 to m = 9,

Solution:

$f(m) = 5 (1.07)^m$

m = 0

=> f(0) = 5(1.07)⁰  = 5  cm

the length of the fish was approximately 9.19 cm.

=>  $9.19 = 5 (1.07)^m$

=>  $1.838 = (1.07)^m$

=> m = 9

Domain = [0 , 9 ]

reasonable domain to plot the growth function = [0 , 9 ]

y-intercept of the graph of the function f(m) represent represent initial length of fish = 5 cm

average rate of change of the function f(m) from m = 1 to m = 9

m = 1  => f(1) = 5(1.07) = 5.35  cm

m = 9 => f(9) = 5(1.07)⁹ = 9.19 cm

Change in 9 - 1 = 8 months = 9.19 - 5.35  =  3.84 cm

average rate of change of the function f(m) from m = 1 to m = 9,  

= (3.84/8)

=  0.48  cm per month

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