Mr. Visca’s: PreCalculus (Calc 3.2)
Unit 3 – Day 2 Exponential Functions and
Modeling
Exponential Population Model:
If a population P is changing at a constant percentage rate r each year, then
___________________________
Where P0 is the initial population, r is expressed as a decimal and t is the time in years.
Does this model look familiar?
𝑃(𝑡) = 𝑃𝑜 (1 ± 𝑟)𝑡
======
f(x) = abx
If r > 0, then P(t) is an exponential growth function, and its growth factor is _________.
If r < 0, then P(t) is an exponential decay function and its decay factor is __________.
Example 1:
Tell whether the population model is growing or decaying and find the constant percentage rate
of growth or decay.
a. San Jose: P(t) = 782,248(1.0136)t
b. Detroit: P(t) = 1,203,368(0.9858)t
c. Rochester: ?
Do you think Rochester’s population model is growing or decaying? Explain why?
Why do you think San Jose’s population is growing and Detroit’s is shrinking?
What is the biggest driver of population growth?
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Other Exponential Growth and Decay Models:
Population growth and decay models are used to represent populations of animals, bacteria and
radioactive atoms. Exponential growth models can be built in terms of the time it takes to
double a quantity or the time it takes to half a quantity.
Model Bacteria Growth
Example 2:
Suppose a culture of 100 bacteria is put into a petri dish and the culture doubles every hour.
Predict when the number of bacteria will reach 350,000.
B(t) = ?
On the flip side, exponential decay models can be built in terms of the time it takes to half a
quantity. Can you name a common exponential decay process where a quantity decays by half?
The time it takes a radioactive process to decay is the half-life.
Model Radioactive Decay
Example 3:
Suppose the half-life of a certain radioactive substance, let’s call it Viscanium, is 20 days and
there are 5 grams present initially. Find the time when there will be 1 gram of the substance
left.
V(t) = ?
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So far our models have been given to us or developed algebraically. However, we could use
exponential regression to build our models from population data.
Example 4: Use the Population data below and graphing calculator to create an exponential
regression equation to predict the population for 2003.
Year
Population (in millions)
1900
76.2
1910
92.2
1920
106.0
1930
123.2
1940
132.2
1950
151.3
1960
179.3
1970
203.3
1980
226.5
1990
248.7
2000
281.4
2003
?
Homework:
section 3.2: #s 2-20 (evens), 30-34 (evens)
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