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
