In Exercises 4 and 5, identify a, b, and x. Then write the exponential function that models the situation. Finally, use the function to answer the question.

4. You invest $2000 in an investment that earns 6% interest, compounded quarterly. How much will the investment be worth after 5 years?

5. You invest $3000 in an investment that earns 5% interest, compounded monthly. How much will the investment be worth after 8 years?

6. The formula that financial managers and accountants use to determine the value of investments that are subject to compounding interest is A = P(1 + r/n)^nt where A is the final balance, P is the initial deposit, r is the annual interest rate, n is the number of times the interest is compounded per year and t is the number of years. Redo Exercises 4 and 5 using this formula.

1 answer

To address your request, we'll follow the steps outlined in Exercises 4 and 5 by identifying \( a \), \( b \), and \( x \) for both situations, writing the exponential functions, and then using those functions to calculate the final balances.

Exercise 4

  1. Identifying values for Exercise 4:

    • \( P = 2000 \): Initial investment
    • \( r = 0.06 \): Annual interest rate (6%)
    • \( n = 4 \): Number of times interest is compounded per year (quarterly)
    • \( t = 5 \): Number of years
  2. Exponential function: The formula for compound interest is: \[ A = P \left( 1 + \frac{r}{n} \right)^{nt} \]

    Plugging in the identified values: \[ A = 2000 \left( 1 + \frac{0.06}{4} \right)^{4 \times 5} \] Simplifying: \[ A = 2000 \left( 1 + 0.015 \right)^{20} \] \[ A = 2000 (1.015)^{20} \]

  3. Calculating the final amount: \[ A \approx 2000 \times 1.346855007 = 2693.71 \]

    Final amount: After 5 years, the investment will be worth approximately $2693.71.


Exercise 5

  1. Identifying values for Exercise 5:

    • \( P = 3000 \): Initial investment
    • \( r = 0.05 \): Annual interest rate (5%)
    • \( n = 12 \): Number of times interest is compounded per year (monthly)
    • \( t = 8 \): Number of years
  2. Exponential function: \[ A = P \left( 1 + \frac{r}{n} \right)^{nt} \]

    Plugging in the identified values: \[ A = 3000 \left( 1 + \frac{0.05}{12} \right)^{12 \times 8} \] Simplifying: \[ A = 3000 \left( 1 + 0.0041667 \right)^{96} \] \[ A = 3000 (1.0041667)^{96} \]

  3. Calculating the final amount: \[ A \approx 3000 \times 1.432364654 = 4297.09 \]

    Final amount: After 8 years, the investment will be worth approximately $4297.09.


Summary of Results

  • In Exercise 4, the final investment value after 5 years is $2693.71.
  • In Exercise 5, the final investment value after 8 years is $4297.09.