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) = 10,125(1.83)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) 9,638 18,794.10 36,648.50 71,464.58
Which account recorded a greater percentage change in amount of money over the previous year? Justify your answer. (5 points)
Part A: The amount of money in account A is increasing. We know this because the expression (1.83)x is being multiplied by 10,125. When x increases, or as the years go by, the multiplier (1.83)x also increases. It increases by a percentage of 83% per year because the base of the exponential, 1.83, is equal to 1 + 0.83 or 100% + 83%.
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 percentage change of A and B from year 1 to year 2.
For account A, let's find f(1) and f(2):
f(1) = 10,125(1.83)^1 = 18,528.75
f(2) = 10,125(1.83)^2 = 33,907.68
Percentage change in account A from year 1 to year 2 = ((f(2) - f(1)) / f(1)) * 100
= ((33,907.68 - 18,528.75) / 18,528.75) * 100 ≈ 82.4 %
For account B, let's find g(1) and g(2).
g(1) = 9,638
g(2) = 18,794.10
Percentage change in account B from year 1 to year 2 = ((g(2) - g(1)) / g(1)) * 100
= ((18,794.10 - 9,638) / 9,638) * 100 ≈ 94.9 %
Thus, account B recorded a greater percentage change in the amount of money over the previous year (94.9% compared to 82.4% in account A).