To determine how many years the engagement will last, we can use the formula for compound interest, which is:
\[ A = P(1 + r)^t \]
Where:
- \(A\) is the future value of the investment (the amount they want to have).
- \(P\) is the principal amount (the initial amount of money, which is the current savings).
- \(r\) is the annual interest rate (as a decimal).
- \(t\) is the number of years the money is invested.
In this case:
- \(A = 57,250\)
- \(P = 24,500\)
- \(r = 8% = 0.08\)
We want to solve for \(t\):
\[ 57,250 = 24,500(1 + 0.08)^t \]
First, we divide both sides by 24,500:
\[ \frac{57,250}{24,500} = (1.08)^t \]
Calculating the left side:
\[ \frac{57,250}{24,500} \approx 2.3333 \]
So we have:
\[ 2.3333 = (1.08)^t \]
To solve for \(t\), we take the logarithm of both sides:
\[ \log(2.3333) = \log((1.08)^t) \]
Using the power rule of logarithms:
\[ \log(2.3333) = t \cdot \log(1.08) \]
Now we can solve for \(t\):
\[ t = \frac{\log(2.3333)}{\log(1.08)} \]
Using a calculator, we find:
\[ \log(2.3333) \approx 0.3700 \] \[ \log(1.08) \approx 0.0334 \]
Now, substituting these values:
\[ t = \frac{0.3700}{0.0334} \approx 11.08 \]
Thus, the engagement will last approximately 11 years.