Questions: [OCR content] Exam #3 Question 10 of 18 (1 point) Question Attempt 1 of 1 Time Remaining: 7 8 9 10 11 12 13 14 15 16 Find the future value of the annuity. Round your answer to the nearest cent. Do not round intermediate steps. Payment Rate Compounded Time 4,000 1.66% Semiannually 3 years The future value of the annuity is Continue

 [OCR content]

Exam #3
Question 10 of 18 (1 point)  Question Attempt 1 of 1

Time Remaining:

7 8 9 10 11 12 13 14 15 16

Find the future value of the annuity. Round your answer to the nearest cent. Do not round intermediate steps.

Payment Rate Compounded Time
4,000 1.66% Semiannually 3 years

The future value of the annuity is 

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Transcript text: [OCR content] Exam #3 Question 10 of 18 (1 point) | Question Attempt 1 of 1 Time Remaining: 7 8 9 10 11 12 13 14 15 16 Find the future value of the annuity. Round your answer to the nearest cent. Do not round intermediate steps. Payment Rate Compounded Time $4,000 1.66% Semiannually 3 years The future value of the annuity is $ Continue
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Solution

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Solution Steps

To find the future value of an annuity, we use the future value of an annuity formula:

FV=P×((1+r)n1r) FV = P \times \left( \frac{(1 + r)^n - 1}{r} \right)

where:

  • P P is the payment amount per period,
  • r r is the interest rate per period,
  • n n is the total number of periods.

Given that the interest is compounded semiannually, we need to adjust the interest rate and the number of periods accordingly.

Step 1: Given Values

We are given the following values for the annuity:

  • Payment per period, P=4000 P = 4000
  • Annual interest rate, r=0.0166 r = 0.0166
  • Compounding periods per year, m=2 m = 2 (semiannually)
  • Total time in years, t=3 t = 3
Step 2: Calculate Rate per Period and Total Periods

The interest rate per period is calculated as: rp=rm=0.01662=0.0083 r_p = \frac{r}{m} = \frac{0.0166}{2} = 0.0083 The total number of periods is: n=m×t=2×3=6 n = m \times t = 2 \times 3 = 6

Step 3: Calculate Future Value of the Annuity

Using the future value of an annuity formula: FV=P×((1+rp)n1rp) FV = P \times \left( \frac{(1 + r_p)^n - 1}{r_p} \right) Substituting the values: FV=4000×((1+0.0083)610.0083) FV = 4000 \times \left( \frac{(1 + 0.0083)^6 - 1}{0.0083} \right) Calculating the expression gives: FV=4000×((1.0083)610.0083)24503.55 FV = 4000 \times \left( \frac{(1.0083)^6 - 1}{0.0083} \right) \approx 24503.55

Final Answer

The future value of the annuity is \\(\boxed{24503.55}\\).

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