Problem 18 - Algebra 2 and Trigonometry

Question

a. When Kyle was born, his grandparents invested \(\$ 5,000\) in a college fund that paid 4\(\%\) per year, compounded yearly. What was the value of this investment when Kyle was ready for college at age 18\(?\) (Note that \(r=0.04 . )\) b. If Kyle's grandparents had invested the \(\$ 5,000\) in a fund that paid 4\(\%\) compounded continuously, what would have been the value of the fund after 18 years?

Step-by-Step Solution

Answer

a. $10,121.86; b. $10,199.14.

Step 1: Understanding Compound Interest

In this problem, we are working with compound interest. Part (a) involves yearly compounding, while part (b) involves continuous compounding. Each has a different formula to calculate the future value of the investment.

Step 1: Calculate Future Value with Yearly Compounding

For part (a), we use the formula for compound interest: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]where \(A\) is the amount of money accumulated after n years, including interest, \(P\) is the principal amount ($5,000), \(r\) is the annual interest rate (0.04), \(n\) is the number of times that interest is compounded per year (1 for yearly), and \(t\) is the number of years the money is invested (18). Plugging in the values: \[ A = 5000 \left(1 + \frac{0.04}{1}\right)^{1 \times 18} = 5000 \times (1.04)^{18} \]Calculate this to find \(A\).

Step 2: Calculate Future Value with Continuous Compounding

For part (b), we use the formula for continuous compounding: \[ A = Pe^{rt} \]where \(A\) is the future value, \(P\) is the principal ($5,000), \(r\) is the annual interest rate (0.04), and \(t\) is the time in years (18). Plugging in the values: \[ A = 5000 \cdot e^{0.04 \cdot 18} \]Calculate this to determine the future value \(A\) when compounded continuously.

Key Concepts

Yearly Compounding

Yearly compounding is a method of calculating interest where the interest is added to the initial principal at the end of each year. This type of compounding is straightforward and is often used in savings accounts and investments. For an investment with yearly compounding, you apply the compound interest formula:

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

In this formula:

• \( A \) is the future value of the investment, including interest.
• \( P \) is the principal amount initially invested.
• \( r \) is the annual interest rate.
• \( n \) is the number of times the interest is compounded per year. For yearly compounding, \( n = 1 \).
• \( t \) represents the investment period in years.

With yearly compounding, you can see the growth of your investment annually. As interest is applied only once per year, this method offers a clear and measurable appreciation of the investment over time.

Continuous Compounding

Continuous compounding is a form of compounding interest where the interest is calculated instantly and repeatedly added to the principal. It represents the case where interest is compounded an infinite number of times per year, which allows for the maximum potential growth of an investment. For continuous compounding, the formula is slightly different:

• \[ A = Pe^{rt} \]

In this formula:

• \( A \) is the amount of money accumulated after time \( t \), with interest.
• \( P \) is the principal investment.
• \( r \) is the annual interest rate.
• \( t \) is the time in years.
• \( e \) is the base of the natural logarithm, approximately equal to 2.71828.

Continuous compounding is a theoretical construct allowing us to understand the idea of exponential growth. It is often used in mathematical finance to model scenarios where variables change instantaneously and continuously.

Future Value Calculation

Future value calculation is the process of determining the worth of an investment after a specified period, taking into account the compounding effects of interest. This future value differs significantly based on whether it is calculated using yearly or continuous compounding.For yearly compounding:

• The formula is \( A = P \left(1 + \frac{r}{n}\right)^{nt} \).
• This calculates an increment in investment once per year based on the interest rate.

For continuous compounding:

• The formula is \( A = Pe^{rt} \).
• This provides a more exponential growth since interest is continuously calculated and added.

The choice of compounding method affects the final future value of an investment. Yearly compounding leads to a more gradual increase, while continuous compounding maximizes growth, theoretically offering higher returns.