EXPLANATION:
Given;
We are told that a deposit of $2000 is invested in an account that yields an annual interest of 4.5%, but is compounded monthly.
Required;
We are required to calculate how much money will be in the investment in 6 years time.
Step-by-step solution.
The formula for calculating a compound interest is given as;
[tex]A=P(1+r)^t[/tex]The variables in this formula are;
[tex]\begin{gathered} A=amount\text{ }after\text{ }the\text{ }period\text{ }given \\ P=principal\text{ }amount\text{ }at\text{ }the\text{ }beginning \\ r=annual\text{ }rate\text{ }of\text{ }interest \\ t=period\text{ }of\text{ }investment\text{ }in\text{ }years \end{gathered}[/tex]However, when the compounding is done at several periods within a year (for example, monthly, quarterly, half-yearly, etc), the formula is amended to reflect the period of compounding within each year.
The amended formula is what we have been given in this question.
Therefore, from the details provided,
[tex]\begin{gathered} P=2000 \\ r=0.045 \\ t=6 \\ n=12 \end{gathered}[/tex]Please note that n is the number of times compounding takes place per year and in this case its 12 times (monthly).
Therefore;
[tex]A=2000(1+\frac{0.045}{12})^{12\times6}[/tex][tex]\begin{gathered} A=2000(1+0.00375)^{72} \\ \\ A=2000(1.00375)^{72} \\ \\ A=2000\times1.30930310151 \\ \\ A=2618.60620301 \end{gathered}[/tex]Rounded to the nearest hundredth, we now have
ANSWER:
[tex]A=2,618.61[/tex]