Question

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)

Answer 1

Answer:

Account A: Decreasing at 8 % per year

Account B: Decreasing at 10.00 % per year

Account B shows the greater percentage change  

Step-by-step explanation:

Part A: Percent change from exponential formula

f(x) = 9628(0.92)ˣ

The general formula for an exponential function is

y = ab^x, where

b = the base of the exponential function.

if b < 1, we have an exponential decay function.

ƒ(x) decreases as x increases.

Account A is decreasing each year.

We can rewrite the formula for an exponential decay function as:

     y = a(1 – b)ˣ, where

1 – b = the decay factor

     b = the percent change in decimal form

If we compare the two formulas, we find

0.92 = 1 - b

b = 1 - 0.92 = 0.08  = 8 %

The account is decreasing at an annual rate of 8 %.

Part B.  Percent change from table

$\begin{array}{cccc}\textbf{r/a} & \textbf{g(r)/\ } & \textbf{Change/\ } & \textbf{\% Change}\\1 & 8972.00 & & \\ & &897.20 &-10.00 \\2 & 8074.80 & & \\ & &807.48 &-10.00\\3 & 7267.32 & & \\ & &726.73 &-10.00 \\4 & 6540.59 & & \\\end{array}$

The account is decreasing at an annual rate of 10.00 %.

Account B recorded a greater percentage change in the amount of money over the previous year.