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.
Thanks for asking, Ms. Sue, however, I do not see why Jordan didn't reply.
Anyway, here's my work for Task 1 of this portfolio:
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.
I chose the number 3.
Number of Hours Number of Bacteria
0 1
1 3
2 9
3 27
4 81
5 243
6 729
After 24 hours, there will be 282,429,536,481 bacteria
Explain why this table represents exponential growth.
This table represents exponential growth because the number of bacteria is always multiplied by 3.
Using this example, explain why any nonzero number raised to a power of zero is equal to one.
The reason any nonzero number raised to power of zero equals one is that any number to the zero power is the product of no numbers, which is multiplicative identity 1.
Write a rule for this table.
y=1×3^x
Suppose you started with 100 bacteria, but they still grew by the same growth factor. How would your rule change? Explain your answer.
I would change the rule into y=100×3^x.
My first rule started from 1 bacterium, and would multiply by 3 for x times. Now that I am starting with 100 bacteria but still using the same growth factor (3x), I would change 1 from my earlier rule into 100, but keep the remaining values same.
I send my work in separate docs. I got 100% in this one, my teacher is waiting for the rest of the documents. I do not think the same system goes for you, but me and my teacher, Mrs. Claire (I'm pretty sure you have the same teacher!) had a "pact" on this. If I get good feedback on this, I will definitely get back. If I do not, you can tell that I did not get a grade 97%+. So, sorry in advance if that happens!
Does anyone have the answer to task two??? Because I’ve been stuck on it for three days
thats not funny bro.
who's gonna take that seriously
Can someone post their answers for task 2 please?
a. To find a city that has experienced population growth, you can conduct research online or refer to reliable sources like census data or population reports. Let's say we find a city called City A, which has experienced population growth. To determine its population on January 1st of a certain year, you would need to access the population data for that city from reliable sources for the specific year you are interested in.
Let's assume the population of City A on January 1st of a certain year is represented by y, and the number of years that have passed since then is represented by x. To write an exponential function that represents the city's population growth, we can use the general form of an exponential function:
y = a * (b^x)
In this equation:
- y represents the population of City A after x years.
- a represents the initial population of City A (population on January 1st of the specific year).
- b represents the growth factor, which determines the rate of population growth over time.
You would need to find specific numerical values for a and b based on the population data for City A and the specific year you're considering.
b. To find a city that experienced population decline during the same time, you can conduct research similar to step a. Let's say we find a city called City B, which had a larger population than City A initially but experienced population decline. Determine the population of City B for January 1st of the same year picked for City A using reliable sources.
Let's assume the population of City B on January 1st of the same year is represented by y, and the number of years that have passed since then is represented by x. To write an exponential function representing the city's population decline, we can modify the equation from part a:
y = a * (b^(-x))
In this equation:
- y represents the population of City B after x years.
- a represents the initial population of City B (population on January 1st of the specific year).
- b represents the decay factor, which determines the rate of population decline over time.
Again, you would need to find specific numerical values for a and b based on the population data for City B and the specific year you're considering.
c. The exponential functions in parts (a) and (b) are similar in structure but differ in the nature of their growth or decline.
In part (a), the exponential function represents population growth, where the population increases exponentially over time. The base, b, is greater than 1, indicating exponential growth.
In part (b), the exponential function represents population decline, where the population decreases exponentially over time. The base, b, is between 0 and 1, indicating exponential decay.
The main similarity between the equations is the general form of an exponential function – y = a * (b^x) or y = a * (b^(-x)). The variables and numbers used are similar, such as y representing the population, x representing the number of years, and a representing the initial population on January 1st of the specific year.
d. To determine when the population of City A first exceeds that of City B, you need to solve the equations for both cities and find when the population of City A becomes greater than City B. Let's assume you have the equations:
For City A: y = a * (b^x)
For City B: y = c * (d^(-x))
To find when the population of City A first exceeds that of City B, you need to solve the following equation:
a * (b^x) > c * (d^(-x))
This equation compares the populations of City A and City B for different values of x (years). You would need to solve the equation algebraically or with the help of a numerical method (such as trial and error) to find the exact year when City A's population exceeds City B's population.
e. To determine when the population of City A will be at least twice the size of the population of City B, you need to solve the equations for both cities and find when the population of City A becomes greater than or equal to twice the population of City B. Let's assume you have the same equations as mentioned in part d.
To find when the population of City A is at least twice the size of City B, you need to solve the following equation:
a * (b^x) >= 2 * (c * (d^(-x)))
This equation compares the populations of City A and City B, making sure that City A's population is at least twice that of City B for different values of x (years).
You would need to solve the equation algebraically or with the help of a numerical method to find the exact year when City A's population is at least twice the population of City B.