Problem 365 - Calculus Volume 1

Question

True or False? If true, prove it. If false, find the true answer. 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

False. You need to invest approximately \(\$6767.12\) initially.

Step 1: Understand the Problem

We need to save \(\$50,000\) in 20 years in an account that offers continuous compounding at 10% interest. The task is to figure out the initial investment required to achieve this.

Step 2: Write the Formula for Continuous Compounding

For continuous compounding, we use the formula \(A = Pe^{rt}\). Here, \(A\) is the amount in the future, \(P\) is the initial principal (the value we need to find), \(r\) is the interest rate, and \(t\) is the time in years.

Step 3: Substitute Known Values into the Formula

Substitute \(A = 50,000\), \(r = 0.10\), and \(t = 20\) into the formula: \[50000 = P \times e^{0.10 \times 20}\].

Step 4: Simplify the Equation

Simplify the expression in the equation: \[e^{0.10 \times 20} = e^2\]. The value of \(e^2\) is approximately 7.389.

Step 5: Solve for P

Rearrange the equation to solve for \(P\): \[P = \frac{50000}{7.389}\].

Step 6: Calculate the Initial Investment

Calculate \(P\) using the approximation:\[P \approx \frac{50000}{7.389} \approx 6767.12\].

Key Concepts

Initial Investment Calculation

When planning to save for future expenses, it is crucial to accurately calculate the initial investment. This is the amount of money you need to deposit now to achieve a specific financial goal at a later date. In our scenario, you want to ensure you have \(50,000 in 20 years for college tuition, assuming a constant 10% interest rate with continuous compounding.To find the initial amount required, the formula for continuous compounding can be rearranged to solve for the principal amount, or initial investment, denoted as \(P\):

• Formula: \(A = Pe^{rt}\)
• A rearranges to \(P = \frac{A}{e^{rt}}\)

where \(A\) is your target amount (in this case, \)50,000), \(r\) is the interest rate (10% or 0.10 as a decimal), and \(t\) is the time period in years (20). Knowing these values, you can calculate the initial investment necessary to meet your goal, ensuring you have enough funds when needed.

Interest Rate

Understanding interest rates is fundamental to calculating investments. An interest rate is the cost of borrowing money or the reward for saving. Continuous compounding, in particular, assumes that the interest is added to the principal constantly, leading to exponential growth over time.For a rate of 10%:

• Write it as a decimal: 0.10
• Combine with the time period for greater clarity: \(e^{0.10 \times 20}\)

This method differs from simple or annual compounding, as it introduces more frequent growth intervals, continuously compounding the interest. Such an arrangement maximizes returns, compounding interest exponentially rather than linearly. Thus, while the rate remains a numerical value, it's the foundation that drives the growth potential of the investment.

Exponential Growth

Exponential growth occurs when each increment in time leads to a proportionally larger increase in the accumulated amount. It is a fundamental concept in situations involving continuous compounding.Using the formula for continuous compounding:

• Each year, your investment grows by a factor of \(e^{rt}\)
• With \(r = 10\%\) and \(t = 20\) years, we have \(e^{2}\), or roughly 7.389 times the initial amount
This effect means that even modest initial investments can grow significantly over time, given a stable environment and interest rate. The power of exponential growth illustrates how compound interest can accelerate savings, making it an attractive method for wealth accumulation over long periods. This highlights the importance of starting investments early, allowing time to maximize growth.