Question 1 (Essay Worth 10 points)
(03.03 MC)
A biologist is studying the growth of a particular species of algae. She writes the following equation to show the radius of the algae, f(d), in mm, after d days:
f(d) = 9(1.04)d
Part A: When the biologist concluded her study, the radius of the algae was approximately 12.81 mm. What is a reasonable domain to plot the growth function? (4 points)
Part B: What does the y-intercept of the graph of the function f(d) represent? (2 points)
Part C: What is the average rate of change of the function f(d) from d = 3 to d = 9, and what does it represent? (4 points)
Question 2 (Essay Worth 10 points)
(03.03, 03.04 MC)
The price of fuel may increase due to demand and decrease due to overproduction. Marco is studying the change in the price of two types of fuel, A and B, over time.
The price f(x), in dollars, of fuel A after x months is represented by the function below:
f(x) = 2.96(1.04)x
Part A: Is the price of fuel A increasing or decreasing and by what percentage per month? Justify your answer. (5 points)
Part B: The table below shows the price g(m), in dollars, of fuel B after m months:
m (number of months) 1 2 3 4
g(m) (price in dollars) 3.04 3.22 3.41 3.61
Which type of fuel recorded a greater percentage change in price over the previous month? Justify your answer. (5 points)
Question 3 (Essay Worth 10 points)
(03.05 HC)
Belinda wants to invest $1,000. The table below shows the value of her investment under two different options for three different years:
Number of years 1 2 3
Option 1 (amount in dollars) 1100 1200 1300
Option 2 (amount in dollars) 1100 1210 1331
Part A: What type of function, linear or exponential, can be used to describe the value of the investment after a fixed number of years using option 1 and option 2? Explain your answer. (2 points)
Part B: Write one function for each option to describe the value of the investment f(n), in dollars, after n years. (4 points)
Part C: Belinda wants to invest in an option that would help to increase her investment value by the greatest amount in 20 years. Will there be any significant difference in the value of Belinda's investment after 20 years if she uses option 2 over option 1? Explain your answer, and show the investment value after 20 years for each option. (4 points)
Question 4 (Essay Worth 10 points)
(03.05 MC)
Aurora is planning to participate in an event at her school's field day that requires her to complete tasks at various stations in the fastest time possible. To prepare for the event, she is practicing and keeping track of her time to complete each station.
The x-coordinate is the station number, and the y-coordinate is the time in minutes since the start of the race that she completed the task.
(1, 3), (2, 6), (3, 12), (4, 24)
Part A: Is this data modeling a linear function or an exponential function? Explain your answer. (2 points)
Part B: Write a function to represent the data. Show your work. (4 points)
Part C: Determine the average rate of change between station 2 and station 4. Show your work. (4 points)
answer each question seperately and in 4 sentences each
Answer for Question 1:
Part A: A reasonable domain to plot the growth function would be all real numbers, as the algae can continue to grow indefinitely.
Part B: The y-intercept of the graph of the function represents the initial radius of the algae when d = 0. In this case, it would be f(0) = 9(1.04)^0 = 9.
Part C: To find the average rate of change from d = 3 to d = 9, we calculate f(9) - f(3) / 9 - 3. Plugging in the values, we get (9(1.04)^9 - 9(1.04)^3) / (9 - 3). This represents the average increase in radius per day from day 3 to day 9.
redo this in a 5th graders vocabulary
Answer for Question 1:
Part A: A reasonable range of days to plot the growth of the algae would be from 0 to a very large number of days, because the algae can keep growing for a long time.
Part B: The y-intercept on the graph of the function represents the radius of the algae when there are 0 days. In this case, it would be 9 mm.
Part C: To find the average rate of change from day 3 to day 9, we calculate how much the radius of the algae increases in total during that time. It's like finding the average increase in size per day from day 3 to day 9.
The price of fuel may increase due to demand and decrease due to overproduction. Marco is studying the change in the price of two types of fuel, A and B, over time.
The price f(x), in dollars, of fuel A after x months is represented by the function below:
f(x) = 2.96(1.04)x
Part A: Is the price of fuel A increasing or decreasing and by what percentage per month? Justify your answer. (5 points)
Part B: The table below shows the price g(m), in dollars, of fuel B after m months:
m (number of months) 1 2 3 4
g(m) (price in dollars) 3.04 3.22 3.41 3.61
Which type of fuel recorded a greater percentage change in price over the previous month? Justify your answer. (5 points)
do this in a 5th graders vocabulary
Answer for Question 2:
Part A: The price of fuel A is increasing. Each month, the price goes up by 4%. This means that it becomes 4% more expensive every month compared to the previous month.
Part B: Looking at the table, we can see that the price of fuel B is also increasing each month, but we need to compare the percentage change. To do this, we can calculate the percentage change between each month. For example, the percentage change from month 1 to month 2 is (3.22 - 3.04) / 3.04 * 100 = 5.92%. We can do the same for the other months. By comparing these percentage changes, we can determine which fuel recorded a greater percentage change in price over the previous month.
Belinda wants to invest $1,000. The table below shows the value of her investment under two different options for three different years:
Number of years 1 2 3
Option 1 (amount in dollars) 1100 1200 1300
Option 2 (amount in dollars) 1100 1210 1331
Part A: What type of function, linear or exponential, can be used to describe the value of the investment after a fixed number of years using option 1 and option 2? Explain your answer. (2 points)
Part B: Write one function for each option to describe the value of the investment f(n), in dollars, after n years. (4 points)
Part C: Belinda wants to invest in an option that would help to increase her investment value by the greatest amount in 20 years. Will there be any significant difference in the value of Belinda's investment after 20 years if she uses option 2 over option 1? Explain your answer, and show the investment value after 20 years for each option. (4 points)
do this in a 5th graders vocabulary
Answer for Question 3:
Part A: Both options for Belinda's investment have values that increase over time. Option 1 has a steady increase of $100 each year, which can be described as a linear function. Option 2 has an increasing rate of growth, where the amount increases more and more each year. This can be described as an exponential function.
Part B: For option 1, the function can be written as f(n) = 100n + 1000, where n represents the number of years. For option 2, the function can be written as f(n) = 1000(1.1)^n, where 1.1 represents the growth rate per year.
Part C: After 20 years, the value of Belinda's investment with option 1 would be $3000 (100(20) + 1000), while the value with option 2 would be approximately $6727 (1000(1.1)^20). Therefore, the value of Belinda's investment after 20 years would be significantly higher if she chooses option 2 over option 1.
Aurora is planning to participate in an event at her school's field day that requires her to complete tasks at various stations in the fastest time possible. To prepare for the event, she is practicing and keeping track of her time to complete each station.
The x-coordinate is the station number, and the y-coordinate is the time in minutes since the start of the race that she completed the task.
(1, 3), (2, 6), (3, 12), (4, 24)
Part A: Is this data modeling a linear function or an exponential function? Explain your answer. (2 points)
Part B: Write a function to represent the data. Show your work. (4 points)
Part C: Determine the average rate of change between station 2 and station 4. Show your work. (4 points)
do this in a 5th graders vocabulary
Answer for Question 4:
Part A: The data is modeling an exponential function. This is because the time it takes to complete each task is doubling as the station number increases. For example, from station 1 to station 2, the time doubles from 3 minutes to 6 minutes. This pattern continues as we move to station 3 and 4.
Part B: The function to represent the data can be written as y = 3 * (2^x), where x represents the station number and y represents the time in minutes.
Part C: To find the average rate of change between station 2 and station 4, we calculate the difference in time and divide it by the difference in station numbers. From station 2 to station 4, the time difference is 24 - 6 = 18 minutes, and the station number difference is 4 - 2 = 2. Therefore, the average rate of change is 18/2 = 9 minutes per station. This means that on average, it takes 9 minutes longer to complete each additional station.