in 9th grade work and writing 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.
Task 1
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 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 create a table of values showing the number of bacteria present after each hour for the first six hours, we need to use the hourly growth rate selected. Let's assume the growth rate is 3.
Hour | Number of Bacteria
------|------------------
1 | 3
2 | 3 x 3 = 9
3 | 9 x 3 = 27
4 | 27 x 3 = 81
5 | 81 x 3 = 243
6 | 243 x 3 = 729
After 24 hours, the number of bacteria present can be calculated by continuing the pattern of exponential growth:
729 x 3 x 3 x 3 = 19683
b. This table represents exponential growth because the number of bacteria is increasing at a constant rate. Each hour, the number of bacteria is multiplied by the growth rate, leading to an exponential increase in the population.
c. In this example, any nonzero number raised to the power of zero is equal to one because raising a number to the power of zero means multiplying it by itself zero times. Since any number multiplied by one is itself, any nonzero number raised to the power of zero is one.
d. The rule for this table can be expressed as: The number of bacteria after each hour is equal to the previous number of bacteria multiplied by the growth rate.
e. If we started with 100 bacteria instead of one, the rule would change to: The number of bacteria after each hour is equal to the previous number of bacteria multiplied by the growth rate, and then the result is added to the initial number of bacteria (100).
Task 2
a. The city chosen for this example is New York City. As of January 1st, 2023, the population of New York City is approximately 8.5 million people. The exponential function representing the city's population is y = 8.5 * (1.015)^x, where y is the population and x is the number of years after 2023. The growth rate of 1.5% is represented by 1.015.
b. The city chosen for this example is Los Angeles. As of January 1st, 2023, the population of Los Angeles is approximately 4 million people. The exponential function representing the city's population is y = 4 * (0.985)^x, where y is the population and x is the number of years after 2023. The decline rate of 1.5% is represented by 0.985.
c. The similarity between the equations in (a) and (b) is that they both represent exponential growth or decline. However, the growth rate in (a) is positive, indicating population growth, while the decline rate in (b) is negative, indicating population decline.
d. To determine the year when the population of New York City first exceeds that of Los Angeles, we need to set the two equations equal to each other and solve for x:
8.5 * (1.015)^x = 4 * (0.985)^x
By solving this equation algebraically or graphically, we can find the value of x, which represents the number of years after 2023. The year can be calculated by adding x to 2023.
e. To determine the year when the population of New York City is at least twice the size of the population of Los Angeles, we need to set up the following inequality:
8.5 * (1.015)^x ≥ 2 * (4 * (0.985)^x)
By solving this inequality algebraically or graphically, we can find the value of x, which represents the number of years after 2023. The year can be calculated by adding x to 2023.
Task 3
a. To model the population of Western Lowland Gorillas after 5, 10, and 20 years, we can use the formula:
y = 360000 * (1 - 0.027)^x, where y represents the population and x represents the number of years since 2022.
After 5 years: y = 360000 * (1 - 0.027)^5
After 10 years: y = 360000 * (1 - 0.027)^10
After 20 years: y = 360000 * (1 - 0.027)^20
b. The table showing the Gorilla population after 5, 10, and 20 years can be constructed using the formula mentioned above:
Years | Population
------|-----------
5 | Calculation
10 | Calculation
20 | Calculation
c. The table shows exponential decay because the population of Western Lowland Gorillas decreases at a constant rate over time. The decay factor of 0.027 represents the annual population decline percentage. Scientists can use exponential decay to predict population changes in endangered species by analyzing the decay factor and the time elapsed. By knowing the initial population and the decline rate, they can estimate the future population and take necessary conservation measures.
In summary, exponential decay models can provide valuable insight into the decline of endangered species' populations, helping scientists understand the severity of the situation and plan conservation efforts accordingly.