To determine the interest rate of the account, we can use the formula for compound interest, which is given by:
\[ A = P(1 + r)^t \]
where:
- \(A\) = the amount of money accumulated after n years, including interest.
- \(P\) = the principal amount (the initial amount of money).
- \(r\) = the annual interest rate (in decimal).
- \(t\) = the time the money is invested for, in years.
In this situation, the account has been deposited with an initial amount of $1,151.00 at \(t_0 = 4.00\) years. The account reaches a value of $1,706.00 at \(t_1 = 19.00\) years. The total time the money is invested is \(t = t_1 - t_0 = 19.00 - 4.00 = 15.00\) years.
So we have:
- \(P = 1,151.00\)
- \(A = 1,706.00\)
- \(t = 15.00\)
Plugging these values into the compound interest formula, we get:
\[ 1,706 = 1,151(1 + r)^{15} \]
Next, we will solve for \(1 + r\):
- Divide both sides by 1,151:
\[ \frac{1,706}{1,151} = (1 + r)^{15} \]
Calculating the left-hand side:
\[ \frac{1,706}{1,151} \approx 1.4805 \]
Now we have:
\[ (1 + r)^{15} \approx 1.4805 \]
- Take the 15th root of both sides:
\[ 1 + r \approx (1.4805)^{\frac{1}{15}} \]
Calculating \( (1.4805)^{\frac{1}{15}} \):
\[ 1 + r \approx 1.0287 \]
- Now solve for \(r\):
\[ r \approx 1.0287 - 1 = 0.0287 \]
- Convert to a percentage:
\[ r \approx 0.0287 \times 100 = 2.87% \]
Thus, the interest rate is approximately 2.87%.