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Math  /  Algebra

QuestionQuestion 2, 10.6.10
HW Score: 0%,00 \%, 0 of 10 points Points: 0 of 1
Use the ordinary annuity formula shown to the right to determine the accumulated amount in the annuity if $80\$ 80 is invested semiannually for 35 year 6.5%6.5 \% compounded semiannually.

Studdy Solution

STEP 1

1. The problem involves an ordinary annuity where payments are made at the end of each period.
2. The payment amount is $80.
3. Payments are made semiannually.
4. The annuity lasts for 35 years.
5. The interest rate is 6.5% compounded semiannually.
6. We need to find the accumulated amount in the annuity.

STEP 2

1. Identify the formula for the ordinary annuity.
2. Define the variables in the formula.
3. Substitute the values into the formula.
4. Calculate the accumulated amount.

STEP 3

Identify the formula for the ordinary annuity. The formula for the future value of an ordinary annuity is:
A=P((1+r)n−1r) A = P \left( \frac{(1 + r)^n - 1}{r} \right)
where: - A A is the accumulated amount. - P P is the payment amount per period. - r r is the interest rate per period. - n n is the total number of periods.

STEP 4

Define the variables in the formula based on the problem statement.
- P=80 P = 80 (since $80 is invested semiannually) - The annual interest rate is 6.5%, so the semiannual interest rate \( r = \frac{6.5\%}{2} = 0.0325 \) - The annuity lasts for 35 years, and since payments are made semiannually, the total number of periods \( n = 35 \times 2 = 70 \)

STEP 5

Substitute the values into the formula:
A=80((1+0.0325)70−10.0325) A = 80 \left( \frac{(1 + 0.0325)^{70} - 1}{0.0325} \right)

STEP 6

Calculate the accumulated amount. First, calculate (1+0.0325)70 (1 + 0.0325)^{70} :
(1+0.0325)70≈10.028 (1 + 0.0325)^{70} \approx 10.028
Now substitute back into the formula:
A=80(10.028−10.0325) A = 80 \left( \frac{10.028 - 1}{0.0325} \right)
A=80(9.0280.0325) A = 80 \left( \frac{9.028}{0.0325} \right)
A=80×277.785 A = 80 \times 277.785
A≈22222.8 A \approx 22222.8
The accumulated amount in the annuity is approximately:
22222.8 \boxed{22222.8}

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