Suppose a certain manufacturer deposits at the beginning of each
3 month period for 6 years in an account paying interest compounded
quarterly. (Round your answers to the nearest cent.)
How much (in ) will be in the account at the end of the 6 year period?
What is the total amount (in ) of interest earned in this account?

Ask by Luna Mitchell. in the United States

Mar 21,2025

Question

Suppose a certain manufacturer deposits at the beginning of each
3 month period for 6 years in an account paying interest compounded
quarterly. (Round your answers to the nearest cent.)
How much (in ) will be in the account at the end of the 6 year period?
What is the total amount (in ) of interest earned in this account?

Ask by Luna Mitchell. in the United States

Mar 21,2025

UpStudy AI · Accepted Answer

We are given that \$7,000 is deposited at the beginning of each quarter for 6 years into an account paying an annual interest rate of 8% compounded quarterly. Since the deposits are made at the beginning of each period, this is an annuity‐due. We will compute the future value (FV) of the annuity and then determine the interest earned.

Let

- $$ P = 7000 $$ (the deposit each period)
- The quarterly interest rate is 
  $$
  r = \frac{8\%}{4} = 0.02.
  $$
- The total number of periods is 
  $$
  n = 6 \text{ years} \times 4 = 24.
  $$

The future value of an annuity due is given by the formula
$$
FV = P \cdot \frac{(1 + r)^n - 1}{r} \cdot (1 + r).
$$

Substituting in our values we have
$$
FV = 7000 \cdot \frac{(1.02)^{24} - 1}{0.02} \cdot 1.02.
$$

**Step 1. Compute $$(1.02)^{24}$$:**

Using the exponential:
$$
(1.02)^{24} \approx 1.608438.
$$

**Step 2. Compute the annuity factor:**
$$
\frac{(1.02)^{24} - 1}{0.02} = \frac{1.608438 - 1}{0.02} \approx \frac{0.608438}{0.02} \approx 30.4219.
$$

Then, multiply by the additional factor $$(1 + r) = 1.02$$:
$$
30.4219 \times 1.02 \approx 31.0303.
$$

**Step 3. Compute the total future value:**
$$
FV \approx 7000 \times 31.0303 \approx 217212.10.
$$

Rounded to the nearest cent, the amount in the account at the end of 6 years is approximately 
$$
\$217,\!212.10.
$$

**Step 4. Compute the total interest earned.**

The total amount deposited is 
$$
7000 \times 24 = \$168,\!000.
$$

Thus, the interest earned is
$$
\text{Interest} = FV - \text{Total Deposits} \approx 217212.10 - 168000 \approx \$49,\!212.10.
$$

**Final Answers:**

- The amount in the account at the end of 6 years is approximately $$\$217,\!212.10$$.
- The total interest earned is approximately $$\$49,\!212.10$$.
At the end of 6 years, there will be approximately \$217,212.10 in the account. The total interest earned is approximately \$49,212.10.

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