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The fast-food restaurant manager receives a $5,000 bonus and adds it to the initial investment. The manager wants to see how much less time it would take the retirement fund to grow with the additional $5,000 and continuously compounded interest. The amount of time it would take the manager to reach a balance of $65,000 with continuously compounded interest can be modeled by the equation ln 3.25 = ln e0.0595t. How much less time does it take the new investment to reach the same balance? Solve for the time in years, rounded to the nearest tenth of a year, showing all steps.

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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.

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