Calculate how much time it will take for a \( \$ 1000 \) investment to grow to at least \( \$ 3,000 \) when it earns a \( 2.4 \% \) annual interest rate. Round your answer to the nearest whole year when the value will be at least \( \$ 3,000 \). It will take about years.

To determine how long it will take for your \$1000 investment to grow to at least \$3000 with a 2.4% annual interest rate, you can use the formula for compound interest: \[ A = P(1 + r)^t \] where \( A \) is the amount of money accumulated after n years, including interest, \( P \) is the principal amount (the initial amount), \( r \) is the annual interest rate (decimal), and \( t \) is the number of years the money is invested. Plugging in your values: \[ 3000 = 1000(1 + 0.024)^t \] This simplifies to: \[ 3 = (1.024)^t \] To solve for \( t \), take the logarithm of both sides: \[ \log(3) = t \cdot \log(1.024) \] Now, you can isolate \( t \): \[ t = \frac{\log(3)}{\log(1.024)} \] Calculating this gives \( t \approx \frac{0.4771}{0.0103} \approx 46.25 \). Rounding to the nearest whole year, it will take about 46 years for the investment to grow to at least \$3000.