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. Usingthisexample,explainwhyanynonzeronumberraisedtoapowerof
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.
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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.
Task 1:
a. Let's suppose the hourly growth rate of the bacteria is 3.
Number of hours | Number of bacteria
-----------------------------------
0 | 1
1 | 3
2 | 9
3 | 27
4 | 81
5 | 243
6 | 729
Using the growth rate of 3, we can determine the number of bacteria after 24 hours by multiplying the initial number of bacteria (1) by the growth rate raised to the power of the number of hours passed. Therefore, we can calculate the number of bacteria after 24 hours as 1 * (3^24) = 1 * (3,486,784,401) = 3,486,784,401.
b. This table represents exponential growth because the number of bacteria is increasing at a constant rate of 3 times each hour. Each subsequent value in the table is the product of the previous value and the growth rate.
c. Any non-zero number raised to the power of zero is equal to one because any number multiplied by one is equal to itself. In the context of exponential growth, when the number of hours is zero, the growth rate is not applied, and the starting value remains unchanged. Therefore, any non-zero number raised to the power of zero equals one due to this property of multiplication.
d. The rule for this table is that the number of bacteria after x hours can be calculated using the formula: Number of bacteria = Initial number of bacteria * (Growth rate) ^ x.
e. If we started with 100 bacteria instead of 1, the rule would change as follows: Number of bacteria = 100 * (Growth rate) ^ x. The initial number of bacteria would be 100 instead of 1, but the growth factor and the rest of the formula remain the same.
Task 2:
a. Let's consider a city called City A. On January 1st of a certain year, City A had a population of 500,000. We can 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.
The equation would be: y = 500,000 * (1.03)^x, where 1.03 represents a 3% growth rate each year.
b. Let's consider a city called City B. On January 1st of the same year, City B had a population of 1,000,000. We can 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.
The equation would be: y = 1,000,000 * (0.95)^x, where 0.95 represents a 5% decline rate each year.
c. The similarity between the equations in parts (a) and (b) is that they both represent exponential growth or decline. In both equations, the population is multiplied by a fixed rate each year. The difference is that the growth rate in equation (a) is greater than 1, resulting in population growth, while the decline rate in equation (b) is less than 1, resulting in population decline.
d. To find the year when the population of City A first exceeds that of City B, we need to set the two equations equal to each other and solve for x:
500,000 * (1.03)^x = 1,000,000 * (0.95)^x
Dividing both sides by 500,000:
(1.03)^x = 2 * (0.95)^x
Next, take the natural logarithm of both sides:
ln((1.03)^x) = ln(2 * (0.95)^x)
Using the power rule of logarithms:
x * ln(1.03) = ln(2) + x * ln(0.95)
Rearranging the equation to isolate x:
x * ln(1.03) - x * ln(0.95) = ln(2)
Factoring out x:
x * (ln(1.03) - ln(0.95)) = ln(2)
Dividing both sides by (ln(1.03) - ln(0.95)):
x = ln(2) / (ln(1.03) - ln(0.95))
Using a calculator, we can find x ≈ 33.78. Therefore, the population of City A will first exceed that of City B in approximately 34 years.
e. To find the year when the population of City A is at least twice the size of the population of City B, we need to solve the following inequality:
500,000 * (1.03)^x ≥ 2 * (1,000,000 * (0.95)^x)
Dividing both sides by 500,000:
(1.03)^x ≥ 2 * (0.95)^x
Taking the natural logarithm of both sides:
ln((1.03)^x) ≥ ln(2 * (0.95)^x)
Using the power rule of logarithms:
x * ln(1.03) ≥ ln(2) + x * ln(0.95)
Rearranging the inequality to isolate x:
x * ln(1.03) - x * ln(0.95) ≥ ln(2)
Factoring out x:
x * (ln(1.03) - ln(0.95)) ≥ ln(2)
Dividing both sides by (ln(1.03) - ln(0.95)):
x ≥ ln(2) / (ln(1.03) - ln(0.95))
Using a calculator, we can find x ≥ 40.79. Therefore, the population of City A will be at least twice the size of the population of City B in approximately 41 years.
Task 3:
a. To model the population of Western Lowland Gorillas after 5, 10, and 20 years, we need to calculate the population using the annual decline rate of 2.7%.
After 5 years:
y = 360,000 * (1 - 0.027)^5
y ≈ 360,000 * (0.973)^5 ≈ 338,309
After 10 years:
y = 360,000 * (1 - 0.027)^10
y ≈ 360,000 * (0.973)^10 ≈ 317,688
After 20 years:
y = 360,000 * (1 - 0.027)^20
y ≈ 360,000 * (0.973)^20 ≈ 272,941
b. The table showing the Gorilla population after 5, 10, and 20 years would look like this:
Number of years (x) | Population (y)
-----------------------------------
5 | 338,309
10 | 317,688
20 | 272,941
c. The table shows exponential decay because the population is decreasing at a constant rate of 2.7% each year. Each subsequent value in the table is the product of the previous value and the decline rate. Scientists can use exponential decay to predict population changes in endangered species by studying the decline rate, analyzing factors contributing to the decline, and applying the decay model to estimate future population sizes. This allows them to assess the impact of human activities and make informed decisions to protect and conserve these species.
In summary, exponential decay can be used to predict population changes in endangered species by mathematically modeling the decline rate and using it to calculate future population sizes. This information helps scientists and conservationists identify critical periods, plan interventions, and develop strategies to prevent extinction and promote the recovery of endangered species.