Answer: To find the time it would take the new investment to reach a balance of $65,000 with continuously compounded interest, we can use the equation:
ln(3.25) = ln(e^(0.0595t)).
First, we need to solve for t. To do that, we can get rid of the natural logarithm on both sides by using the property: ln(e^x) = x.
So, the equation becomes:
3.25 = e^(0.0595t).
Now, let's isolate t:
Take the natural logarithm on both sides:
ln(3.25) = ln(e^(0.0595t)).
Using the property: ln(e^x) = x, we get:
ln(3.25) = 0.0595t.
Now, divide by 0.0595 to solve for t:
t = ln(3.25) / 0.0595.
Let's calculate the value of t:
t = ln(3.25) / 0.0595 ≈ 35.81 years.
So, it would take approximately 35.81 years for the new investment to reach a balance of $65,000 with continuously compounded interest.
Now, let's calculate how much less time it would take with the additional $5,000.
Let's assume the original time it would take to reach $65,000 without the $5,000 bonus is t_1, and the time it would take with the $5,000 bonus is t_2.
t_1 = 35.81 years (from the previous calculation).
The new investment has an additional $5,000, so the future value (FV) of the investment will be $65,000 + $5,000 = $70,000.
Using the same formula as before, we have:
ln(3.25) = ln(e^(0.0595 * t_2)).
Now, solve for t_2:
3.25 = e^(0.0595 * t_2).
Take the natural logarithm on both sides:
ln(3.25) = 0.0595 * t_2.
Now, divide by 0.0595 to solve for t_2:
t_2 = ln(3.25) / 0.0595 ≈ 36.92 years.
So, with the additional $5,000 bonus, it would take approximately 36.92 years to reach a balance of $70,000 with continuously compounded interest.
Now, to find how much less time it would take, we subtract t_1 from t_2:
Time difference = t_2 - t_1 ≈ 36.92 - 35.81 ≈ 1.11 years.
Therefore, the new investment with the $5,000 bonus takes approximately 1.11 years less to reach a balance of $65,000 compared to the original investment.