Alg2_M5_5.10
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Nova Southeastern University *
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Mathematics
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Feb 20, 2024
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Module 5, Exponential & Logistic Growth Assignment
In this module you have learned about exponential growth and decay. These types of models can
be used for situations in which the increase or decrease in a population becomes more rapid as
time passes. On paper, any population can grow or decay exponentially forever, but in the world
outside the mathematics classroom there are often restrictions that cause the growth or decay to
slow as time passes. This type of growth or decay is referred to logistic.
Since 2009 the number of smartphones shipped from manufacturers
to stores around the world has increased exponentially. The growth
from 2009 through 2015 can be modeled using the function
G
(
t
)=
174
•
1.67
t
where
t
is the number of years since 2009 and
G
(
t
)
is measured in millions of smartphones.
Use this information and your knowledge of exponential and logistic
growth to explore the mathematical concepts in the questions below.
1. Create a graph of the function
G
(
t
)
and explain the meaning of the y-intercept in terms of
the number of smartphones being shipped to stores.
Answer
: The y-intercept occurs when t = 0,
representing the initial year (2009). G(0) =
174 million smartphones, indicating the
starting point of the exponential growth.
2. Explain a way you could calculate exactly when smartphone manufacturers were shipping 500
million smartphones to stores around the world.
Answer
: When G(t) = 500 million
We can convert the exponential function into its logarithmic form to solve for t:
Formula:
If y = xn, then logxy = n
Solution:
G(t) = 174(1.67t)
500 = 174(1.67t)
500/174 = 1.67t
log1.67(500/174) = t
t ≈ 2.058 years
Therefore year = 2009 + 2.058
The year ≈ 2011
3. Use your plan from question 2 to find when smartphone manufacturers were shipping 750
million smartphones to stores around the world. If you are solving algebraically, explain which
properties of logarithms you used and why.
Answer
: When G(t) = 750 million
Solution:
G(t) = 174(1.67t)
750 = 174(1.67t)
750/174 = 1.67t
log1.67(750/174) = t
t ≈ 2.849 years
Therefore year = 2009 + 2.849
Year ≈ 2012
4. Find the values
G
(
7
)
and
G
(
8
)
. Show your work and explain what these numbers
mean in context of this scenario.
Answer
: G(t) = 174(1.67)^t
G(7) = 174(1.67)^7
= 6303.25 million phones are shipped to stores
G(8) = 174(1.67)^8
=10526.43 million phones are shipped to stores
The answers indicate the amount of smartphones shipped during the years of 2016 and 2017.
5. Suppose that the actual number of smartphones (in millions) shipped from manufacturers to
stores in 2016 was 1,584 and the actual number in 2017 was 1,651. Does the model
G
(
t
)=
174
•
1.67
t
accurately reflect this growth? Explain why the given model is returning
values that are so much higher than the actual numbers.
Answer
: The model G(t)=174⋅1.67^t does not accurately reflect the actual
numbers’ growth because, based on the percentage of the increased growth,
they should be the same, but the thing is if I get the percentage increase for
every year in 2016 to 2017’s actual numbers it is just 4% increase but if I
refer to the model G(t)=174⋅1.67^t its 36% and I calculated this for every
year the growth of it increases.
6. Nikola thinks that the model that reflects the growth of smartphones shipped from
manufacturers to stores around the world may be logistic rather than exponential.
Do you agree
with Nikola? Why or why not?
HINT: Consider the differences between exponential and logistic growth, and how these
differences apply to this scenario.
Answer
: I agree with Nikola because when most people have a smartphone, that is, the variable
starts getting closer to its capacity, the demand will start to have a slight decrease, until it
stabilizes.
7. Let the function
F
(
t
)=
1727.93
1
+
9.21
e
−
0.65
t
reflect the growth of the number of smartphones
shipped from manufacturers to stores around the world, where
t
is the number of years since
2009 and
F
(
t
)
is measured in millions. Find the year when the number of smartphones
shipped around the world is 1,728. Interpret your answer.
Answer
: The function given is F(t) = 1727.931 + 9.21e^{-0.65t, where t is the number of years
since 2009 and F(t) is measured in millions. We need to find the year when the number of
smartphones shipped is 1,728 million.
Setting F(t) equal to 1,728 and solving for t:
1727.931 + 9.21e^{-0.65t}=1728
9.21e^{-0.65t} =0.069
Now, isolate the exponential term:
e^{-0.65t} = 0.069/9.21
e^{-0.65t}≈0.0075
Take the natural logarithm (ln) of both sides:
-0.65t ≈ln(0.0075)
Now, solve for t:
t ≈ ln(0.0075)/-0.65
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Using a calculator, we find:
t ≈15.45
So, the year when the number of smartphones shipped is approximately
1,728 million is 2009 + 15.45 ≈2024.45.
Therefore, the interpretation is that around the year 2024, according to this logistic model, the
number of smartphones shipped globally is expected to reach 1,728 million.