Problem 13 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the accumulated amount \(A\) if the principal \(P\) is invested at the interest rate of \(r /\) year for \(t\) yr. $$ P=\$ 2500, r=7 \%, t=10, \text { compounded semiannually } $$

Step-by-Step Solution

Answer

The accumulated amount, A, after investing $2500 at a 7% annual interest rate for 10 years compounded semiannually, is approximately $3536.52.

Step 1: Identify the formula for compound interest

In this case, since the investment is compounded semiannually, we will use the compound interest formula with semiannual compounding: \(A = P(1 + \frac{r}{2n})^{nt}\) where: A = the accumulated amount after t years P = the principal amount invested r = the annual interest rate (in decimal form) n = the number of compounding periods per year t = the number of years

Step 2: Convert the interest rate to decimal form

The given interest rate is 7%, which we should convert to a decimal by dividing by 100: \(r = \frac{7}{100} = 0.07\)

Step 3: Substitute the given values into the formula

Now, we will substitute the given values of P, r, n, and t into the formula: \(A = 2500(1 + \frac{0.07}{2 \cdot 2})^{2 \cdot 10}\)

Step 4: Evaluate the expression

Simplify the expression and calculate the accumulated amount: \(A = 2500(1 + \frac{0.07}{4})^{20}\) \(A = 2500(1.0175)^{20}\) \(A ≈ 2500 \cdot 1.414609\) \(A ≈ 3536.52\)

Step 5: Interpret the result

The accumulated amount, A, after investing \(2500 at a 7% annual interest rate for 10 years compounded semiannually, is approximately \)3536.52.

Key Concepts

Financial Mathematics

Understanding financial mathematics is essential for making informed decisions about investments, savings, and loans. It revolves around the concept of time value of money, which states that the value of money changes over time due to interest or returns. One of the key principles in this field is compound interest, which, unlike simple interest, calculates interest on both the initial principal and the accumulated interest from previous periods.

This allows the sum to grow at an exponential rate, which is particularly important when considering long-term investments or retirement savings. In the compound interest formula, factors such as the principal amount \( P \), the interest rate \( r \), the number of compounding periods \( n \), and the total number of periods \( t \) play crucial roles. By meticulously considering each of these variables, one can forecast the future value of an investment.

Accumulated Amount Calculation

The accumulation of wealth over time is often the goal when investing money. To calculate the accumulated amount in a compound interest scenario, a specific formula is used, which incorporates the frequency of compounding.

For instance, when compounding semiannually, interest is added to the principal twice a year. This frequency is crucial to determining the final amount, as it influences how often interest is compounded and thus how quickly the investment grows. The formula for the accumulated amount with compound interest is \( A = P(1 + \frac{r}{n})^{nt} \), where \( A \) represents the future value of the investment after \( t \) years. By substituting known values into this formula, such as the initial investment (the principal), the interest rate (in decimal form), the number of compounding periods per year, and the duration of the investment, one can calculate the future worth of their assets.

Exponential Functions

Exponential functions are the mathematical backbone of compound interest calculations. These functions model situations where a quantity grows at a rate proportional to its current value, resulting in an exponential increase over time. The compound interest formula \( A = P(1 + \frac{r}{n})^{nt} \) is a classic example of an exponential function, where the amount \( A \) increases exponentially as time \( t \) passes.

The power of exponential growth becomes evident as we observe how investments compound over years, dramatically enhancing their value, especially as \( t \) increases. By understanding the nature of exponential functions, students can better grasp the potential of long-term investments and the significance of compounding frequencies, such as semiannual compounding in the given example. Knowledge of exponential functions ensures a more profound appreciation of how time and rate can dramatically impact the growth of an investment.