Question

The amount of money in an account may increase due to rising stock prices and decrease due to falling stock prices. Mason 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.

Please help me

Answer 1

Concept:

A part or quantity in every hundred is defined as a percentage. It's a fraction having a denominator of 100. The percentage difference is the change in the value of a quantity over time expressed as a percentage difference. Percentage Change is a term used to describe the growth or decrease in a quantity expressed in percentages.

Find:

Whether the amount is increasing or decreasing over the years and which is experiencing a greater change.

Solution:

Part A:

From the given equation, $1.83 > 1$, this shows that money in account A is $83 \%$ more, compared to its previous year. The amount will be increasing by $83 \%$.

Part B:

The change in each consecutive term is $94 \%$. This shows that the consecutive term is multiplied by $94 \%$  the previous year's amount. As a result, Part B's account has a higher percentage change in money over the preceding year than Part A's account.

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Answer 2

Answer:

From part A, the percentage change in the amount of money in account A over the previous year exists 9%.

As a consequence, account B recorded a greater percentage change in the amount of money over the previous year.

Step-by-step explanation:

Exponential Growth and Decay Function:

A function that expands and shrinks at a constant percentage rate is named an exponential growth function and an exponential decay function, respectively.

1. Exponential growth functions are of the form$E(x)=a(1+r)^{x}$, where $0 < r < 1$ denotes the percent change in decimal form.

2. Exponential decay functions are of the form $E(x)=a(1-r)^{x}$, where $0 < r < 1$ denotes the percent change in decimal form.

Step 1

Rewrite the given function $f(x)=1264(1.09)^{x}$

$f(x) &=1264(1.09)^{x}$

$&=1264(1+0.09)^{x}$

Compare the function f with the general form of the exponential growth function $E(x)=a(1+r)^{x}$and determine the values of a and r.

$&a=1264$

$&r=0.09$

Since$r > 0$, the function is increasing.

Convert the value of $r$ into percentage form.

$r &=0.09$

$&=9 \%$

As a result, the amount of money in account A increases at the rate of$9 \%$ per year.

Step 2

The percent change is calculated using the formula $\frac{\text { New value }-\text { Old Value }}{\text { Old Value }} \times 100$

Calculate the percentage change in the amount of money in account B in the second year.

$\text { Percent Change } &=\frac{g(2)-g(1)}{g(1)} \times 100$

$&=\frac{1512.50-1375}{1375} \times 100$

$&=\frac{137.5}{1375} \times 100$

$&=10$

Compute the percentage change in the amount of money in account B in the third year with respect to the second year.

$\text { Percent Change } &=\frac{g(3)-g(2)}{g(2)} \times 100$

$&=\frac{1663.75-1512.50}{1512.50} \times 100$

$&=\frac{151.25}{1512.50} \times 100$

$&=10$

Calculate the percentage change in the amount of money in account B in the fourth year with respect to the third year.

$\text { Percent Change } &=\frac{g(4)-g(3)}{g(3)} \times 100$

$&=\frac{1830.13-1663.75}{1663.75} \times 100$

$&=\frac{166.38}{1663.75} \times 100$

$& \approx 10$

From the above computations, it can be observed that the percentage change in the amount of money in account B over the previous year is around 10%

From part A, the percentage change in the amount of money in account A over the previous year exists 9 %.

As a consequence, account B recorded a greater percentage change in the amount of money over the previous year.

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