Answer:
8 years and 3.54 months
Step-by-step explanation:
If $10,000 is put aside in a money market account with interest compounded monthly at 2.2%, we can find the time required for the account to earn $2000 by using the Compound Interest Formula:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Compound Interest Formula}}\\\\A=P\left(1+\dfrac{r}{n}\right)^{nt}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\;\textsf{$A$ is the future value.}\\\phantom{ww}\bullet\;\;\textsf{$P$ is the principal amount.}\\\phantom{ww}\bullet\;\;\textsf{$r$ is the interest rate (in decimal form).}\\\phantom{ww}\bullet\;\;\textsf{$n$ is the number of times interest is applied per year.}\\\phantom{ww}\bullet\;\;\textsf{$t$ is the time (in years).}\end{array}}[/tex]
In this case, the future value is the sum of the deposit ($10,000) and the amount earned ($2,000), so:
- A = $10,000 + $2,000 = $12,000
- P = $10,000
- r = 2.2% = 0.022
- n = 12 (monthly)
Substitute these values into the formula and solve for t:
[tex]12000=10000\left(1+\dfrac{0.022}{12}\right)^{12t}[/tex]
[tex]\dfrac{12000}{10000}=\left(1+\dfrac{0.022}{12}\right)^{12t}[/tex]
[tex]1.2=\left(1+\dfrac{0.022}{12}\right)^{12t}[/tex]
[tex]\ln(1.2)=\ln\left(\left(1+\dfrac{0.022}{12}\right)^{12t}\right)[/tex]
[tex]\ln(1.2)=12t\ln\left(1+\dfrac{0.022}{12}\right)[/tex]
[tex]t=\dfrac{\ln(1.2)}{12\ln\left(1+\dfrac{0.022}{12}\right)}[/tex]
[tex]t=8.294937903069...[/tex]
[tex]t = \rm 8\;years \;3.54\; months[/tex]
Therefore, it will take approximately 8 years and 3.54 months for the account to earn $2000.
[tex]\dotfill[/tex]
Additional Notes
If you want the months rounded to the nearest month, it would be approximately 8 years and 4 months.
