First, let's identify the initial amount in the account. The initial amount is [tex]$\$[/tex] 7,595.96$.
Next, we are given that the rate of increase per year is [tex]$2\%$[/tex]. This rate can be expressed as a decimal for calculations, which is [tex]$1.02$[/tex] (since [tex]$2\%$[/tex] increase means multiplying by [tex]$1 + \frac{2}{100} = 1.02$[/tex]).
We need to find the equation that accurately represents the amount of money in the account, [tex]$y$[/tex], after [tex]$x$[/tex] years of continuous growth without any deposits or withdrawals.
The general form for compound interest (where interest is compounded annually) is:
[tex]\[ y = P \times (1 + r)^x \][/tex]
where:
- \( P \) is the principal amount (initial amount),
- \( r \) is the annual growth rate as a decimal,
- \( x \) is the number of years.
Substituting the given values:
- The initial principal amount \( P \) is [tex]$\$[/tex] 7,595.96$,
- The annual growth rate \( r \) is [tex]$0.02$[/tex], which translates to multiplying by [tex]$1.02$[/tex].
Thus, the equation to find \( y \), the amount of money in the account after \( x \) years, is:
[tex]\[ y = 7,595.96 \times (1.02)^x \][/tex]
Therefore, the correct choice is:
[tex]\[ \boxed{y = 7,595.96(1.02)^x} \][/tex]
This matches the third equation listed in the options. Therefore, you should choose:
[tex]\[ y = 7,595.96(1.02)^x \][/tex]
In conclusion, the correct equation is option number 3.
