Problem 365 - Calculus Volume 1

Question

For the following exercises, use \(y=y_{0} e^{k t}\). You are trying to save $$\$ 50,000$$ in 20 years for college tuition for your child. If interest is a continuous \(10 \%,\) how much do you need to invest initially?

Step-by-Step Solution

Answer

Invest approximately $6768 initially.

Step 1: Understand the Equation

The formula \( y = y_0 e^{kt} \) represents continuous compound interest. Here, \( y \) is the final amount, \( y_0 \) is the initial investment, \( k \) is the continuous interest rate, and \( t \) is the time in years.

Step 2: Identify Given Values

From the problem, we have \( y = 50000 \) (the amount needed in 20 years), \( k = 0.10 \) (continuous interest rate of 10%), and \( t = 20 \) years.

Step 3: Rearrange the Equation

To find \( y_0 \), rearrange the equation to solve for initial investment: \( y_0 = \frac{y}{e^{kt}} \).

Step 4: Substitution

Substitute the given values into the rearranged equation: \( y_0 = \frac{50000}{e^{0.10 \times 20}} \).

Step 5: Calculate Exponential Term

Calculate the exponential part: \( e^{0.10 \times 20} = e^{2} \). The value of \( e^2 \) is approximately 7.389.

Step 6: Final Calculation

Now calculate \( y_0 = \frac{50000}{7.389} \), which gives approximately \( y_0 = 6768 \).

Step 7: Conclusion

You need to invest approximately \( 6768 \) dollars initially to grow to \( 50000 \) dollars in 20 years at a continuous interest rate of 10%.

Key Concepts

Exponential Growth

Exponential growth is a phenomenon where a quantity increases at a consistent rate over time. This type of growth occurs in many scenarios, from populations to investments, as in this exercise. With exponential growth, the rate of change is proportional to the current amount, leading to faster increases as the amount grows. In the context of compound interest, your investment's growth accelerates over time because it earns interest not only on the initial amount but also on the accumulated interest.

In the formula \( y = y_0 e^{kt} \), \( y \) represents the future value of the investment, \( y_0 \) is the initial investment, \( e \) is Euler's number (approximately 2.71828), \( k \) is the continuous growth rate, and \( t \) is the time period. Here, the exponential term \( e^{kt} \) signifies compound growth over time.

As you can see, small changes in time \( t \) or growth rate \( k \) can lead to significant changes in the outcome due to this exponential nature. This makes exponential growth a powerful concept in financial planning and other fields. Understanding it helps you make informed choices, especially with long-term investments.

Initial Investment Calculation

Calculating the initial investment required to achieve a specific future value is a core financial concept, crucial for achieving financial goals. In this exercise, we want to determine how much money needs to be invested today to reach \( \\(50,000 \) in 20 years, with a continuous interest rate of 10%.

The formula used is \( y = y_0 e^{kt} \). To find the initial investment \( y_0 \), rearrange the formula to \( y_0 = \frac{y}{e^{kt}} \).

Here’s a step-by-step approach:

• Identify \( y = 50000 \), the future value.
• Determine \( k = 0.10 \), representing a 10% continuous interest rate.
• Set \( t = 20 \), the time period in years.
• Substitute these values into the formula: \( y_0 = \frac{50000}{e^{0.10 \times 20}} \).
• Calculate \( e^{2} \) (since \( 0.10 \times 20 = 2 \)), which is approximately 7.389.
• Finally, divide \( 50000 \) by 7.389, resulting in \( y_0 \approx 6768 \).

This calculation shows that you would need to invest approximately \( \\)6,768 \) initially to achieve your educational savings goal in 20 years.

Financial Planning for Education

Planning for educational expenses can be daunting, yet it is a vital aspect of financial planning. The cost of education is continually rising, making it essential to start saving early. A strategic approach can involve using tools like continuous compound interest to grow your savings effectively.

When you plan for education, consider these key factors:

• Research the projected costs of tuition and associated educational expenses for your timeframe.
• Determine the amount you would like to save and when you will need it.
• Choose an investment strategy that fits your risk tolerance and timeline. Continuous compound interest is a good option if you have a long-term horizon.
• Regularly review your investment plan to ensure it remains aligned with your educational financial goals.

By calculating and understanding your initial investment needs, as illustrated by the exercise, you can set realistic saving goals. This foresight helps protect against future financial burdens and supports achieving your child's educational aspirations. Developing a comprehensive plan empowers you to make steady progress towards these goals.