Problem 40 - Calculus and its applications

Question

Tania wants to have \(\$ 20,000\) in 5 yr for her dream vacation. Find the continuous money stream, \(R(t)\) dollars per year, that she needs to invest at \(5.125 \%,\) compounded continuously, to generate \(\$ 20,000\).

Step-by-Step Solution

Answer

Tania needs to invest approximately \(3,509.92\) dollars per year continuously.

Step 1: Understand the formula for continuous compounding

For continuous compounding, the future value, \( F \), is given by the formula \( F = R(t) \times \frac{e^{rt}-1}{r} \), where \( R(t) \) is the continuous money stream, \( r \) is the interest rate, and \( t \) is the time in years.

Step 2: Identify the given values

From the problem, we know that \( F = 20,000 \), \( t = 5 \) years, and the interest rate \( r = 0.05125 \) (converted from a percentage to a decimal).

Step 3: Plug in the known values into the formula

Substitute the known values into the equation: \( 20,000 = R(t) \times \frac{e^{0.05125\times5}-1}{0.05125} \).

Step 4: Calculate the exponential term

Calculate \( e^{0.05125\times5} = e^{0.25625} \), which is approximately \( 1.29212 \).

Step 5: Find the fraction term

Subtract 1 from the exponential result to get \( 1.29212 - 1 = 0.29212 \). Divide this by the interest rate: \( \frac{0.29212}{0.05125} \approx 5.69805 \).

Step 6: Solve for the continuous money stream \( R(t) \)

With the equation \( 20,000 = R(t) \times 5.69805 \), solve for \( R(t) \) by dividing both sides of the equation by \( 5.69805 \) to isolate \( R(t) \): \( R(t) \approx \frac{20,000}{5.69805} \).

Step 7: Calculate \( R(t) \)

Perform the division to find \( R(t) \): \( R(t) \approx 3,509.92 \).

Key Concepts

future value calculation

In financial planning, calculating the future value of an investment helps you understand how much you can expect to have after a certain period of time. Here, the future value, denoted as \( F \), represents the amount of money you aim to achieve, given a specific investment plan. For Tania's dream vacation, her future goal is set at \( \$20,000 \).
To reach this future value, you need to determine how much money must be invested today, under continuous compounding conditions. This involves rearranging the future value formula for continuous compounding:

• Make sure to identify the future value, \( F \), which is your target.
• Use the formula: \( F = R(t) \times \frac{e^{rt}-1}{r} \), where \( R(t) \) is the continuous money stream you need to find.

In our scenario: plugging the numbers, you aim to discover \( R(t) \), knowing \( F \) already. This calculation helps in planning how much needs to be saved or invested continuously over time to meet the future monetary requirement.

exponential growth

Exponential growth is a key concept in understanding how investments grow over time under continuous compounding. This growth means that not only the principal amount earns interest but also the accumulating interest does, at every moment.

In Tania's case, the growth is calculated using the expression \( e^{rt} \), where \( e \) is the base of natural logarithms (approximately 2.71828), \( r \) represents the annual interest rate, and \( t \) stands for time in years. Here’s how it works:

• Multiply the interest rate \( r = 0.05125 \) by the time period \( t = 5 \).
• Calculate the exponential function: \( e^{0.05125 \times 5} \), resulting in approximately \( 1.29212 \).

The exponential growth in the formula adjusts how much the investment will accumulate over time. It highlights how a small interest rate over a few years can still lead to significant growth, thanks to the compound nature of \( e^{rt} \).
Understanding this helps appreciate why continuous compounding is often more beneficial than traditional periodic compounding.

interest rate conversion

Converting interest rates is essential when working with different financial calculations. Often, interest rates are given as percentages, and for formulaic calculations, you'll need them in decimal form.
In the formula for continuous compounding, the interest rate \( r \) is used in its decimal form to accurately calculate exponential growth:

• Convert the given percentage \( 5.125\% \) to a decimal: \( 5.125\% = 0.05125 \).

Using these decimal conversions in formulas helps standardize calculations and minimize errors. It’s crucial to understand that the exact value of the interest rate plays a critical role in predicting how much you'll end up with, as small differences can significantly affect the outcome over longer periods of time.
By consistently converting percentages to decimals in your calculations, you increase your accuracy in predicting financial growth through investments.