Problem 10 - Advanced Engineering Mathematics

Question

When interest is compounded continuously, the amount of money increases at a rate proportional to the amount \(S\) present at time \(t\), that is, \(d S / d t=r S\), where \(r\) is the annual rate of interest. (a) Find the amount of money accrued at the end of 5 years when \(\$ 5000\) is deposited in a savings account drawing \(5 \frac{3}{4} \%\) annual interest compounded continuously. (b) In how many years will the initial sum deposited bave doubled? (c) Use a calculator to compare the amount obtained in part (a) with the amount \(S=5000\left(1+\frac{1}{4}(0.0575)\right)^{5(4)}\) that is arcrued when interest is compounded quarterly.

Step-by-Step Solution

Answer

(a) $6665; (b) 12.06 years; (c) $6633 with quarterly compounding.

Step 1: Understand the continuous compounding formula

For continuous compounding, the formula for the future value is given by: \[ S(t) = S_0 e^{rt} \]where \( S_0 \) is the initial amount, \( r \) is the annual rate of interest, \( t \) is the time in years, and \( e \) is the base of the natural logarithm, approximately 2.71828.

Step 2: Calculate the amount after 5 years (Part a)

Given: - \( S_0 = 5000 \) (initial deposit)- \( r = 0.0575 \) (annual interest rate in decimal form)- \( t = 5 \)Substitute these into the continuous compounding formula:\[ S(5) = 5000 \times e^{0.0575 \times 5} \]\[ S(5) = 5000 \times e^{0.2875} \] Using a calculator, \( e^{0.2875} \approx 1.333 \). Thus,\[ S(5) = 5000 \times 1.333 \approx 6665 \] (rounded to the nearest dollar).

Step 3: Determine the time for the amount to double (Part b)

We want \( S(t) = 2S_0 \). Using the continuous compounding formula:\[ S(t) = S_0 e^{rt} = 2S_0 \]Dividing both sides by \( S_0 \):\[ e^{rt} = 2 \]Taking the natural logarithm of both sides:\[ rt = \ln(2) \]Solve for \( t \): \[ t = \frac{\ln(2)}{r} \approx \frac{0.6931}{0.0575} \approx 12.06 \] The investment will double in approximately 12.06 years.

Step 4: Compare with quarterly compounding (Part c)

The formula for quarterly compounding is:\[ S = P\left(1 + \frac{r}{n}\right)^{nt} \]Here, \( P = 5000 \), \( r = 0.0575 \), \( n = 4 \), and \( t = 5 \):\[ S = 5000\left(1 + \frac{0.0575}{4}\right)^{4 \times 5} \]\[ S = 5000\left(1 + 0.014375\right)^{20} \]Calculate \( (1.014375)^{20} \approx 1.3266 \), thus:\[ S \approx 5000 \times 1.3266 \approx 6633 \] (rounded to the nearest dollar).Compare this with \( 6665 \) from continuous compounding.

Key Concepts

Interest Rate Calculation

Interest rate calculation is essential when determining how much your investment will grow over time. By calculating the interest rate, you can understand your returns better and make informed financial decisions. There are many types of interest rate calculations, but continuous compounding is particularly exciting. This method increases your money exponentially, allowing your investment to grow faster compared to simple or periodic compounding.
To calculate the future value with continuous compounding, use the formula:

• \( S(t) = S_0 e^{rt} \)

Here, \( S_0 \) is the initial principal amount, \( e \) is the base of natural logarithms (approx. 2.71828), \( r \) is the annual interest rate, and \( t \) is the time in years. Calculating with this formula enables you to see how much your initial investment grows in the future.
Understanding and using the interest rate calculation effectively is key to maximizing your investment returns, especially when it comes to continuous compounding.

Exponential Growth

Exponential growth is a powerful concept in finance and investments. It's when the value of an investment grows at a rate proportional to its current value, resulting in the value increasing rapidly over time. This is precisely what happens with continuous compounding.
With continuous compounding, the investment grows exponentially. For instance, using the formula for continuous compounding \( S(t) = S_0 e^{rt} \), each small increment of time increases the investment slightly, and these small increases accumulate over time to create significant growth.
This concept is critical in scenarios where you are looking for long-term investment growth. The beauty of exponential growth lies in its potential to turn a modest investment into a substantially larger amount over time. By understanding exponential growth, investors can make strategic decisions, benefiting from the enhanced rate of return due to continuous compounding.

Quarterly Compounding

Quarterly compounding is a common method of calculating interest that gives investors another way to grow their savings. Unlike continuous compounding, which assumes that interest is added constantly, quarterly compounding adds interest four times a year.
Using the formula for quarterly compounding:

• \( S = P\left(1 + \frac{r}{n}\right)^{nt} \)

Where \( P \) is the principal amount, \( r \) is the annual interest rate, \( n \) is the number of compounding periods per year (in this case, 4), and \( t \) is the number of years.
Quarterly compounding has the advantage of being more frequent than annual compounding but less so than continuous compounding. This can offer a good balance, giving investors the benefit of compounding without needing continuous calculations. Comparing the results of continuous vs. quarterly compounding helps demonstrate their differences in returns over time.

Natural Logarithm

The natural logarithm, often abbreviated as \( \ln \), is an important mathematical function used extensively in continuous compounding interest calculations. It helps solve problems where the rate of growth depends on the current value, such as finding out how long it takes for an investment to double.
In finance, the natural logarithm is used to calculate the time it will take for an investment to reach a certain value. For example, to determine how long it will take for your money to double with continuous compounding, you would rearrange the formula \( S(t) = S_0 e^{rt} \) to solve for \( t \).Taking the natural logarithm of both sides, you would use the formula:

• \( t = \frac{\ln(2)}{r} \)

Here, \( \ln(2) \) represents the time it takes for the investment to double assuming \( r \) remains constant. This is a crucial tool for investors wanting to forecast growth over time.