3 years ago, you had [tex]$120 in the bank. You now have $[/tex]192. If the amount of money at the end of each year increases exponentially, how much will you have in the bank after another 7 years assuming that you do not deposit or withdraw any money?

A. [tex]$307.20
B. $[/tex]491.52
C. [tex]$590.20
D. $[/tex]574.89



Answer :

Let's break down the problem step by step to find the solution:

1. Initial Data:
- 3 years ago, the initial amount of money was [tex]$120. - The current amount of money is $[/tex]192.
- A total of 3 years have passed.

2. Annual Growth Rate:
We need to find the annual growth rate, which can be determined using the formula for exponential growth:
[tex]\[ \text{Annual Growth Rate} (r) = \left(\frac{\text{Current Amount}}{\text{Initial Amount}}\right)^{\frac{1}{\text{Number of Years}}} \][/tex]
Plugging in the values:
[tex]\[ r = \left(\frac{192}{120}\right)^{\frac{1}{3}} \][/tex]
After calculating, the annual growth rate is approximately 1.1696.

3. Projecting Future Amount:
We need to project the amount of money in the bank after another 7 years, making it a total of 10 years from the initial amount of [tex]$120. The amount of money after a certain number of years with exponential growth is given by: \[ \text{Future Amount} = \text{Current Amount} \times (r)^{\text{Years to Project}} \] Where the current amount is $[/tex]192, [tex]\( r \)[/tex] is the annual growth rate we calculated, and the years to project is 7:
[tex]\[ \text{Future Amount} = 192 \times (1.1696)^7 \][/tex]
After evaluating this expression, the future amount is approximately [tex]$574.89. Therefore, the amount you will have in the bank after another 7 years is $[/tex]574.89, which matches one of the given options. So, the answer is:

$574.89