Final answer:
To find out how long it will take for Megan's investment to reach $45,000 at an 8% interest rate compounded monthly, we use the compound interest formula. After rearranging the formula and solving for the unknown time period, we calculate the result using the properties of logarithms.
Step-by-step explanation:
To determine how long it will take for Trust Officer Megan to grow an investment of $15,000 at an interest rate of 8% compounded monthly to reach a value of $45,000, we can use the compound interest formula:
A = P(1 + \frac{r}{n})^{nt}
Where:
- A is the future value of the investment/loan, including interest.
- P is the principal investment amount ($15,000).
- r is the annual interest rate (decimal).
- n is the number of times that interest is compounded per year.
- t is the number of years the money is invested.
We want to find t when:
A = $45,000
P = $15,000
r = 0.08
n = 12
Solve for t:
$45,000 = $15,000(1 + \frac{0.08}{12})^{12t}
Divide both sides by $15,000 and simplify to get:
3 = (1 + \frac{0.08}{12})^{12t}
Take the natural logarithm of both sides:
ln(3) = ln((1 + \frac{0.08}{12})^{12t})
ln(3) = 12t * ln(1 + \frac{0.08}{12})
Now, solve for t:
t = \frac{ln(3)}{12 * ln(1 + \frac{0.08}{12})}
When you calculate that, you get the number of years needed for the account value to reach $45,000.