A = [tex]35 \times \frac{\left[\left(1+\frac{0.0725}{12}\right)^{(12 \cdot 42)}-1\right]}{\left(\frac{0.0725}{12}\right)}[/tex]
A =
(Round to the nearest cent as needed.)
Answer (2)
Calculate the value inside the parenthesis: $1 +
\frac{0.0725}{12}
\approx 1.00604167$.
Calculate the exponent: $(12
\cdot 42) = 504$.
Calculate the value of $\left(1+\frac{0.0725}{12}\right)^{(12
\cdot 42)} \approx 20.81842568$.
The final answer, rounded to the nearest cent, is 114810.19 .
Explanation
Understanding the Formula We are given the formula for calculating the future value of an annuity: A = 35 × ( 12 0.0725 ) [ ( 1 + 12 0.0725 ) ( 12 ⋅ 42 ) − 1 ] Our goal is to calculate the value of A and round it to the nearest cent.
Performing the Calculations First, we need to calculate the value inside the innermost parentheses: 1 + 12 0.0725 = 1 + 0.006041666... ≈ 1.00604167 Next, we calculate the exponent: 12 × 42 = 504 Now, we raise the result from the first step to the power of the result from the second step: ( 1.00604167 ) 504 ≈ 20.81842568 Subtract 1 from this result: 20.81842568 − 1 = 19.81842568 Divide by 12 0.0725 :
12 0.0725 = 0.006041666... ≈ 0.00604167 0.00604167 19.81842568 ≈ 3280.291147 Finally, multiply by 35: 35 × 3280.291147 ≈ 114810.1901 Rounding to the nearest cent, we get $114810.19.
Final Result Therefore, the value of A rounded to the nearest cent is $$114810.19.
Examples
Annuities are commonly used in retirement planning. For example, if you deposit $$35 per month into an account with an annual interest rate of 7.25% compounded monthly for 42 years, the annuity formula helps you calculate the future value of your investment. This calculation is crucial for estimating how much money you'll have saved by the end of the investment period, allowing you to plan your retirement finances effectively.
To find the value of A, we first compute the parameters such as monthly interest and number of payments, then use these in the provided formula. After performing the calculations, we determine that A is approximately $114810.19 when rounded to the nearest cent.
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