If anyone does this lol idk but im not gonna look at this because im already done (with the first and second task)
Task 1
Bacteria are the most common example of exponential growth. Research and find a
bacterium that grows each hour exponentially and determine how much it grows
per 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. 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 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
Greece is currently experiencing a financial crisis.
a. Research the financial crisis in Greece and summarize it in one-two
paragraphs.
b. You are in charge. By what percentage will you tell Greece to cut their
spending? What is the decay factor?
c. Write a function modeling this debt situation if the initial debt in 2009 was
$500 billion and using the decay factor found in (b). Let y be measured in
billions of dollar and x represent the number of years since 2009.
d. When will Greece be debt-free if you are in charge? Should you reconsider your answer to (b)?
You tried
Task 1:
a. To determine the number of bacteria present after each hour, you need to know the growth factor - the amount by which the population increases each hour. Once you have the growth factor, you can use it to calculate the number of bacteria for each hour. For example, if the growth factor is 2, it means that the population doubles every hour.
b. The table represents exponential growth because the number of bacteria is increasing at a consistent rate based on the growth factor. Each hour, the number of bacteria is multiplied by the growth factor, resulting in exponential growth.
c. In this example, any number raised to the power of zero is equal to one because when calculating the growth after zero hours, there is no change. Starting with one bacterium and raising it to the power of zero means there are still one bacterium present.
d. The rule for this table would be to raise the growth factor to the power of the number of hours elapsed. For example, if the growth factor is 2 and you want to know the number of bacteria after 3 hours, you would calculate 2^3 = 8. So the rule would be: number of bacteria = growth factor ^ number of hours.
e. If you started with 100 bacteria instead of one, the rule would still be the same. The growth factor determines the rate of growth, so as long as the growth factor remains the same, the rule remains unchanged. The initial population only affects the starting point.
Task 2:
a. To find a city that has experienced population growth, you can research cities known for their increasing population. Once you determine a city, find the population on January 1st of a specific year, and let that be the starting point for your exponential function. The equation should represent population growth as a function of the number of years that have passed.
b. For a city that has experienced population decline, find the population on January 1st of the same year as the previous city. Write an exponential function that represents population decline based on the number of years that have passed.
c. The equations in (a) and (b) will be similar in form but will differ in terms of the growth/decay factor and the initial population. Both equations should have the same independent variable, representing the number of years, but the growth/decay factor and initial population will vary.
d. To determine the year when the population of city (a) exceeds that of city (b), you need to solve the equations for the population of both cities as a function of time. Set the two equations equal to each other and solve for the time (number of years) when the population of city (a) becomes greater than city (b).
e. To find the year when the population of city (a) is at least twice the size of city (b), you can set up an inequality using the two equations. Define a condition where the population of city (a) is greater than or equal to two times the population of city (b). Solve the inequality to find the time (number of years) when this condition is met.
Task 3:
a. To research the financial crisis in Greece, you can look for reliable sources such as news articles, reports from reputable organizations, or government publications. Read and summarize the main causes and effects of the crisis in one to two paragraphs.
b. As the one in charge, you need to decide the percentage by which Greece should cut their spending. Consider the current situation and the economic factors at play. Determine a percentage you believe will help Greece address their financial crisis effectively. The decay factor represents the rate of decline in spending.
c. Write a function to model the debt situation in Greece based on the information you have gathered. Use the initial debt in 2009, which was $500 billion, and the decay factor you calculated in (b). Let y represent the debt measured in billions of dollars, and x represent the number of years since 2009. The function should describe how the debt decreases over time.
d. To find when Greece will be debt-free, you can set the debt function from (c) equal to zero and solve for x (the number of years). This will give you the number of years it will take for Greece's debt to reach zero. Evaluate your answer to (b) to determine if it is realistic or if you need to reconsider your recommended spending cuts.