Question

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?

Short answer

You need to invest approximately $6767.27 initially.

Step-by-step solution

Identify the Given Values

The problem provides us with the final amount needed for college tuition, which is \(y = \$50,000\), the continuous interest rate \(k = 0.10\) (10%), and the time period \(t = 20\) years. We need to find the initial amount \(y_0\) to invest.

Use the Formula

The formula for continuous compounding is given by \( y = y_{0} e^{k t} \). Substitute the known values into the equation: \( 50,000 = y_{0} e^{0.10 \times 20} \).

Simplify the Exponent

Calculate the exponent \(0.10 \times 20\) which equals 2. So the equation becomes \(50,000 = y_{0} e^{2}\).

Solve for Initial Amount \(y_0\)

To find \(y_0\), divide both sides of the equation by \(e^{2}\): \( y_{0} = \frac{50,000}{e^{2}} \).

Calculate \(e^2\)

Calculate \(e^2\) using a calculator, which is approximately 7.389.

Calculate \(y_0\)

Divide \(50,000\) by 7.389 to find \(y_0\): \( y_0 \approx \frac{50,000}{7.389} \approx 6767.27 \).

Key concepts

Exponential Growth

Exponential growth is a powerful concept, often seen in financial contexts such as interest calculations. It describes situations where the increase happens at a rate proportional to the current value. This means that over time, the growth rate itself accelerates. The mathematical expression for exponential growth is usually in the form of the equation \( y = y_0 e^{kt} \). Here,

• \( y \) is the amount you're solving for, usually the future value.
• \( y_0 \) is the initial amount (we'll dive into this concept later).
• \( e \) is the base of the natural logarithm, approximately equal to 2.71828.
• \( k \) is the growth rate (in this case, the interest rate).
• \( t \) is the time period over which the growth happens.

In the given problem, we're looking at savings growing continuously over 20 years. By harnessing exponential growth, we can understand how an initial small amount grows to meet a future financial need like college tuition. This concept is fundamental in tackling investments, savings, and understanding how money compounds over time.

Interest Rate

The interest rate is a crucial element when considering investments, especially in continuous compounding scenarios. It determines the pace of exponential growth in financial contexts. Here, the interest rate is continuously compounded at 10%. This means the rate is applied in an uninterrupted manner over time.
Think of the interest rate \(k\) as a speedometer for your money—higher rates result in faster "speed" or growth of your investment.Some key takeaways about interest rates for continuous compounding:

• A higher rate means faster growth but also entails higher risk in investments.
• Interest rates can vary based on the type of investment or savings account.
• Continuous compounding assumes that the interest is calculated and added to the principal at every moment, theoretically providing the most growth possible for a given rate.

Understanding interest rate mechanics is essential for making educated decisions about where and how much to invest, especially over a long period, as in securing future funds for education.

Initial Investment

The initial investment, represented as \( y_0 \) in our equation, is the amount of money you start with. In financial planning, particularly when targeting a future goal such as paying for education, knowing how much to begin with can define your strategy.Analyzing the given exercise, let's explore the importance of initial investments:

• In the context of \( y = y_0 e^{kt} \), \( y_0 \) is what you invest initially to achieve a target amount \( y \) over time \( t \) with growth rate \( k \).
• An understanding of \( y_0 \) allows for setting realistic financial goals and savings plans.
• Investing more initially can significantly reduce the impact of future shortfalls, compensating for lower interest rates or shorter time periods.

The calculations from the exercise show that to reach \\(50,000 at a 10% continuous rate in 20 years, an initial investment of approximately \\)6767.27 is needed. This emphasizes the power of starting early and the effect of exponential growth in reaching financial milestones.