Answer:
The account will take 7 years to reach a value of $4,020
Step-by-step explanation:
Compound Interest
When it occurs interest in the next period is then earned on the principal sum plus previously accumulated interest.
The formula is:
[tex]{\displaystyle A=P\left(1+{\frac {r}{n}}\right)^{nt}}[/tex]
Where:
A = final amount
P = initial principal balance
r = interest rate
n = number of times interest applied per time period
t = number of time periods elapsed
Ella invested P=$2,700 in an account with an interest rate of r=5.4% (0.054) compounded monthly. Since there are 12 months in a year, n=12.
It will be calculated when will the account have a value of A=$4,020. Substituting all the values in the formula:
[tex]{\displaystyle 4,020=2,700\left(1+{\frac {0.054}{12}}\right)^{12t}}[/tex]
Calculating:
[tex]{\displaystyle 4,020=2,700\left(1.0045)^{12t}}[/tex]
Dividing by 2,700:
[tex]{\displaystyle \frac{4,020}{2,700}=\left(1.0045)^{12t}}[/tex]
To solve this equation for t, we need to apply logarithms:
[tex]{\displaystyle \log\frac{4,020}{2,700}=\log\left(1.0045)^{12t}}[/tex]
Applying the logarithm power rule:
[tex]\displaystyle \log\frac{4,020}{2,700}=12t\log 1.0045[/tex]
Dividing by 12log 1.0045:
[tex]\displaystyle t=\frac{\log\frac{4,020}{2,700}}{12\log 1.0045}[/tex]
Calculating:
t= 7.39 years
Rounding to the nearest year, the account will take 7 years to reach a value of $4,020
