The amount of money in an account may increase due to rising stock prices and decrease due to falling stock prices. Marco is studying the change in the amount of money in two accounts, A and B, over time.
The amount f(x), in dollars, in account A after x years is represented by the function below:
f(x) = 9,628(0.92)x
Part A: Is the amount of money in account A increasing or decreasing and by what percentage per year? Justify your answer. (5 points)
Part B: The table below shows the amount g(r), in dollars, of money in account B after r years:
r (number of years) 1 2 3 4
g(r) (amount in dollars) 8,972 8,074.80 7,267.32 6,540.59
Which account recorded a greater percentage change in amount of money over the previous year? Justify your answer. (5 points)
Part A:
Notice how 0.92<1, which means that the function is decaying at a rate of 8% per year. This means that the amount of money will decrease by 8% per year.
Part B:
Here, we see that the decay rate is 10% per year since each consecutive term is being multiplied by 90% which is 10% less than 100%. Thus, Part B's account experiences a greater percentage change in amount of money over the previous year than Part A's account of 8%.
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Part A:
To determine whether the amount of money in account A is increasing or decreasing and by what percentage per year, we need to analyze the function f(x) = 9,628(0.92)^x.
1. First, note that the base of the exponential function (0.92) is less than 1. This indicates that the amount of money in account A is decreasing with time.
2. To find the percentage change per year, we can subtract the amount in the following year from the amount in the current year, divide the result by the amount in the current year, and then multiply by 100.
Let's calculate the percentage change from year 0 to year 1:
f(1) = 9,628(0.92)^1
f(1) ≈ 9,628(0.92)
f(1) ≈ 8,857.76
Percentage change from year 0 to year 1: ((8,857.76 - 9,628) / 9,628) * 100
≈ -8%
Thus, the amount of money in account A is decreasing by approximately 8% per year.
Part B:
To determine which account recorded a greater percentage change in the amount of money over the previous year, we need to compare the percentages of change between account A and account B.
In Part A, we found that account A has an annual percentage change of -8%.
Now, let's calculate the annual percentage changes for account B using the provided table:
Percentage change from year 1 to year 2: ((8,074.80 - 8,972) / 8,972) * 100 ≈ -10%
Percentage change from year 2 to year 3: ((7,267.32 - 8,074.80) / 8,074.80) * 100 ≈ -10%
Percentage change from year 3 to year 4: ((6,540.59 - 7,267.32) / 7,267.32) * 100 ≈ -10%
Account B also has an annual percentage change of approximately -10% for each year.
Comparing the percentages of change, we can see that account B had a greater decrease in the amount of money over the previous year compared to account A, which had a decrease of 8%.
Therefore, account B recorded a greater percentage change in the amount of money over the previous year.