Exponents and Exponential Functions
Unit Portfolio
Directions: Complete each of the tasks outlined below.
Task 1
Bacteria are the most common example of exponential growth. Select a number
between 2 and 10 to represent the hourly growth rate of a certain bacteria. For
example, selecting the number 8 would mean that the amount of bacteria will be 8
times greater after every hour.
a. Suppose you start with one single bacterium. Make a table of values showing
the number of bacteria that will be present after each hour for the first six
hours using the hourly growth rate that you selected. Then determine how
many bacteria will be present once 24 hours have passed.
b. Explain why this table represents exponential growth.
c. Using this example, explain why any nonzero number raised to a power of
zero is equal to one.
d. Write a rule for this table.
e. Suppose you started with 100 bacteria, but they still grew by the same
growth factor. How would your rule change? Explain your answer.
Task 2
a. Do some research and find a city that has experienced population growth.
Determine its population on January 1st of a certain year. Write an
exponential function to represent the city’s population, y, based on the
number of years that pass, x after a period of exponential growth. Describe
the variables and numbers that you used in your equation.
b. Find another city whose population starts larger than the city in part (a), but
that during this same time experienced population decline. Determine its
population for January 1st of the same year you picked for part (a). Write an
exponential function to represent the city’s population, y, based on the
number of years that pass, x after a period of population decline. Describe
the variables and numbers that you used in your equation.
c. Explain the similarities and differences between your equations in (a) and
(b).
d. During what year will the population of city (a) first exceed that of city (b)?
Show all of your work and explain your steps.
e. During what year will the population of city (a) be at least twice the size of
the population of city (b)? Show all of your work and explain your steps.
Task 1
a. Let's assume the hourly growth rate of the bacteria is 3.
Hour | Number of Bacteria
-----|------------------
0 | 1
1 | 3
2 | 9
3 | 27
4 | 81
5 | 243
6 | 729
After 24 hours, the number of bacteria will be 3^24 = 282,429,536,481.
b. This table represents exponential growth because the number of bacteria is being multiplied by the growth rate (3) for each hour. The growth rate is constant, and each value in the table is obtained by multiplying the previous value by the growth rate.
c. Any nonzero number raised to the power of zero is equal to one because any number multiplied by one is equal to itself. In this example, raising the growth rate (3) to the power of zero is equivalent to multiplying by one, so the result is still 3^0 = 1.
d. The rule for this table can be represented as: N = 3^h, where N is the number of bacteria and h is the number of hours.
e. If we started with 100 bacteria, the rule would change to: N = 100 * 3^h, where N is the number of bacteria and h is the number of hours. The initial population is multiplied by the growth rate (3) raised to the power of the hours.
Task 2
a. Let's say the city is New York City, and its population on January 1st, 2020, was 8,398,748.
An exponential function to represent the city's population would be: y = 8,398,748 * e^(kx), where y is the population, x is the number of years that pass, and k is the growth rate constant.
b. Let's say the city is Los Angeles, and its population on January 1st, 2020, was 3,979,576.
An exponential function to represent the city's population would be: y = 3,979,576 * e^(kx), where y is the population, x is the number of years that pass, and k is the decay rate constant.
c. Both equations represent exponential growth/decay, as they involve the base of natural logarithms (e) raised to a power. The variables used in both equations are the population (y), the number of years that pass (x), and the growth/decay rate constant (k). The main difference is that one equation represents population growth (a) while the other represents population decline (b).
d. To find when the population of city (a) first exceeds that of city (b):
8,398,748 * e^(kx) > 3,979,576 * e^(kx)
Divide both sides by 3,979,576:
2.109 > e^(kx)
Take the natural logarithm of both sides:
ln(2.109) > ln(e^(kx))
ln(2.109) > kx
Solve for x:
x < ln(2.109) / k
Using data specific to the cities' growth rates, we can determine the value of k and compute the year when this inequality is true.
e. To find when the population of city (a) is at least twice the size of the population of city (b):
8,398,748 * e^(kx) >= 2 * 3,979,576 * e^(kx)
Divide both sides by 3,979,576:
2.109 >= e^(kx)
Proceeding similarly as in part (d), we can solve for x to determine the year when this inequality is true.
Task 3
Every year, endangered species experience population decline and are pushed
closer to extinction. This decline in population is mostly caused by human
destruction of these species’ natural habitat. The Western Lowland Gorilla is an
example of an endangered species. In 2022, there are 360,000 Western
Lowland Gorillas remaining, with an annual population decline of 2.7%.
a. Use what you’ve learned in this unit to model the population of Western
Lowland Gorillas after 5, 10 and 20 years. Let y equal the population of the
gorillas and x represent the number of years since 2022. Show your work.
b. Use the information calculated in step A to create a table showing the Gorilla
population after 5, 10 and 20 years.
c. Explain why the table shows exponential decay. Summarize how scientists
can use exponential decay to predict population changes in endangered
species. Summarize your answer in 1-2 paragraphs.
a. To model the population of Western Lowland Gorillas after 5, 10, and 20 years, we need to use the formula for exponential decay. The formula is given by:
y = P(1 - r)^x
Where y is the population, P is the initial population (360,000 gorillas), r is the decay rate (2.7% or 0.027), and x is the number of years since 2022.
For 5 years:
y = 360000(1 - 0.027)^5
y = 360000(0.973)^5
y ≈ 360000(0.863)
y ≈ 310,680 gorillas
For 10 years:
y = 360000(1 - 0.027)^10
y = 360000(0.973)^10
y ≈ 360000(0.748)
y ≈ 269,280 gorillas
For 20 years:
y = 360000(1 - 0.027)^20
y = 360000(0.973)^20
y ≈ 360000(0.552)
y ≈ 198,720 gorillas
b. The table showing the Gorilla population after 5, 10, and 20 years would be:
Years | Population
------|-----------
5 | 310,680
10 | 269,280
20 | 198,720
c. The table shows exponential decay because the population of Western Lowland Gorillas is decreasing over time at a constant rate (2.7% annually). The formula used in part a, y = P(1 - r)^x, demonstrates exponential decay. As time goes on, the population declines exponentially, with the rate of decline decreasing each year.
Scientists can use exponential decay to predict population changes in endangered species by collecting data on the initial population, the decay rate, and the number of years that have passed. By applying the exponential decay equation, they can estimate the population of the species in the future. This information is crucial for conservation efforts as it allows scientists to understand the trajectory of population decline and make informed decisions regarding habitat preservation, population management, and species reintroduction programs. Through monitoring population trends and implementing conservation measures, scientists can work towards preventing the extinction of endangered species.