$A = $80 \times \frac{\left[\left(1+\frac{0.08}{12}\right)^{(12 \cdot 43)}-1\right]}{\left(\frac{0.08}{12}\right)}$
Answer (2)
Calculate the value inside the parenthesis: 1 + 12 0.08 ≈ 1.006666 .
Calculate the exponent: 12 ⋅ 43 = 516 .
Raise the result to the power: ( 1 + 12 0.08 ) 516 ≈ 30.83295 .
Calculate the final value of A: A ≈ 357995.44 .
Explanation
Understanding the Formula We are given the formula: A = 80 × ( 12 0.08 ) [ ( 1 + 12 0.08 ) ( 12 ⋅ 43 ) − 1 ] This formula calculates the future value of a series of payments, which is commonly used in financial mathematics. Here's what each component represents:
A represents the future value of the investment.
$80 represents the periodic payment amount.
0.08 represents the annual interest rate.
12 represents the number of compounding periods per year.
43 represents the number of years.
Objective Our goal is to calculate the value of A using the given formula. We will proceed step by step to ensure accuracy.
Calculations First, we need to calculate the value inside the parenthesis: 1 + 12 0.08 = 1 + 0.006666... ≈ 1.006666 Next, we calculate the exponent: 12 ⋅ 43 = 516 Now, we raise the result from the first step to the power calculated in the second step: ( 1 + 12 0.08 ) ( 12 ⋅ 43 ) = ( 1.006666 ) 516 ≈ 30.83295 Subtract 1 from the result: 30.83295 − 1 = 29.83295 Divide 0.08 by 12 :
12 0.08 = 0.006666... Divide the result from the previous subtraction by the result of the division: 0.006666 29.83295 ≈ 4474.94 Finally, multiply this result by $80 :
80 × 4474.94 ≈ 357995.44
Final Answer Therefore, the value of A is approximately $357995.44 .
Examples
This type of calculation is commonly used when planning for retirement or calculating the future value of an investment account with regular contributions. For example, if you deposit $80 per month into an account that earns 8% annual interest, compounded monthly, over 43 years, this formula tells you how much money you'll have at the end of the 43 years. Understanding compound interest and future value calculations is crucial for making informed financial decisions and planning for long-term financial goals.
The formula calculates the future value of monthly payments of $80 at an annual interest rate of 8% over 43 years. By breaking down the steps of the formula, we find that the total amount accumulated is approximately 357995.44 . This demonstrates the power of compound interest in savings plans.
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