Problem 7 - Applied Calculus

Question

A person deposits money into an account, which pays \(6 \%\) interest compounded continuously, at a rate of \(\$ 1000\) per year for 30 years. Calculate: (a) The balance in the account at the end of the 30 years. (b) The amount of money actually deposited into the account. (c) The interest earned during the 30 years.

Step-by-Step Solution

Answer

The balance is approximately \(\$84160\). Deposited amount is \(\$30000\). Interest earned is \(\$54160\).

Step 1: Identifying the Given Information

The account accrues interest at a rate of \( 6\% \), compounded continuously, with a deposit of \( \$1000 \) per year. The total time span considered is 30 years for this investment problem.

Step 2: Set Up the Formula for Continuous Compounding

The formula for continuous compounding when making regular contributions is derived from the future value of an annuity. The formula is given by \( A = Pe^{rt} \) for continuous compounding, but in this case with regular deposits, it's \( FV = P \times \frac{e^{rt} - 1}{r} \). We need to adjust it for annual deposits.

Step 3: Substitute Known Values into the Formula for (a)

To determine the balance after 30 years, we use the adjusted formula: \[ FV = 1000 \times \frac{e^{0.06 \times 30} - 1}{0.06} \]. Substitute \( P = 1000 \), \( r = 0.06 \), and \( t = 30 \) to compute the balance.

Step 4: Compute the Balance for (a)

Calculate \( e^{0.06 \times 30} \), find the difference, then divide by \( 0.06 \), and multiply by \( 1000 \). This gives us:\[ FV \approx 1000 \times \frac{e^{1.8} - 1}{0.06} \].Using \( e^{1.8} \approx 6.0496 \), we find:\[ FV \approx 1000 \times \frac{6.0496 - 1}{0.06} \approx 1000 \times \frac{5.0496}{0.06} \approx 1000 \times 84.16 \approx 84160. \]

Step 5: Calculate the Amount Deposited for (b)

The total amount deposited into the account over 30 years is simply the annual contribution multiplied by the number of years:\( 1000 \times 30 = 30000. \)

Step 6: Compute the Interest Earned for (c)

The interest earned is found by subtracting the total deposited from the future value:\[ \text{Interest} = 84160 - 30000 = 54160. \]

Key Concepts

Interest Calculation

Interest calculation forms the backbone of any investment or savings plan. When interest is compounded continuously, it means the interest is being calculated and added to the principal balance at every moment. This is beneficial for investors because it allows their investment to grow exponentially over time.

For continuous compounding, the formula is represented by:

• \( A = Pe^{rt} \), where:
• \( A \) is the amount of money accumulated after n years, including interest.
• \( P \) is the principal amount (initial deposit or loan amount).
• \( r \) is the annual interest rate (decimal).
• \( t \) is the time the money is invested for, in years.
• \( e \) is the base of the natural logarithm, approximately equal to 2.7183.

Continuous compounding tends to yield higher returns than periodic compounding because the interest is calculated and added to the principal continuously, effectively giving a slightly higher effective interest rate.

Annuity Future Value

The future value of an annuity is a critical concept when it comes to regular investments or savings plans. An annuity involves making a series of regular payments over time. In our case, it’s the regular yearly deposit of \( \\(1000 \). When interest is compounded continuously, finding the future value involves modifying the basic compounding formula to account for these repeated contributions.
Use the formula:

• \( FV = P \times \frac{e^{rt} - 1}{r} \)
• \( P \) is the regular deposit amount \( \\)1000 \).
• \( r \) is the interest rate \( 0.06 \).
• \( t \) is the total time in years \( 30 \).

This formula helps capture the impact of making regular deposits under continuous compounding conditions, reflecting how investments appreciate over time due to the interest accrued on not only the initial amount but also each subsequent deposit.

Investment Problem

Solving an investment problem involves understanding how much is invested, how it grows, and the interest earned. First, determine the total amount deposited. In our scenario, annually for 30 years:

• Total Deposited = \(1000 \times 30 = \\(30000 \).

Next, assess the investment growth using the future value formula for each regular deposit. For our example:

• Future Value \( FV \approx \\)84160 \).

Lastly, compute the interest earned. Subtract the total amount invested, \( \\(30000 \), from the future value:

• Interest Earned \( = 84160 - 30000 = \\)54160 \).

This exercise exhibits the power of compounding over an extended period, demonstrating how substantial the interest can become relative to the original sum invested. It highlights that the longer the money is invested, particularly with continuous compounding, the more dramatic the growth due to the interest on interest effect.