Math Snap

STEP 1

1. The initial amount of money invested, $A_0$, is $6,000.
2. The final amount, $A(t)$, is double the initial amount, so $A(t) = 2 \times 6,000 = 12,000$.
3. The annual interest rate, $r$, is 2.8%, or 0.028 in decimal form.
4. The interest is compounded continuously.
5. We are using the continuous compound interest formula: $A(t) = A_0 e^{r t}$.
6. We need to find the time $t$ it takes for the investment to double.

STEP 2

1. Set up the equation using the continuous compound interest formula.
2. Solve for $t$ using logarithms.
3. Calculate $t$ and round to two decimal places.

STEP 3

Set up the equation using the continuous compound interest formula:
$$A(t) = A_0 e^{r t}$$ Substitute the known values into the equation:
$$12,000 = 6,000 \times e^{0.028 t}$$

STEP 4

Solve for $t$ using logarithms.
First, divide both sides by 6,000 to isolate the exponential term:
$$2 = e^{0.028 t}$$ Take the natural logarithm of both sides to solve for $t$:
$$\ln(2) = \ln(e^{0.028 t})$$ Using the property of logarithms, $\ln(e^{x}) = x$, we have:
$$\ln(2) = 0.028 t$$

Solve for $t$ by dividing both sides by 0.028:
$$t = \frac{\ln(2)}{0.028}$$ Calculate $t$ using a calculator:
$$t \approx \frac{0.693147}{0.028}$$ $$t \approx 24.76$$ The time it takes for the investment to double is approximately $\boxed{24.76}$ years.