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 Bird Bowers. 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 Bird Bowers. in the United States

Mar 21,2025

UpStudy AI · Accepted Answer

We are given that the manufacturer deposits $$ \$7{,}000 $$ at the beginning of each 3‐month period for 6 years into an account earning $$ 8\% $$ per annum compounded quarterly. Since deposits are made at the beginning of each period, this is an annuity due.

**Step 1. Determine the periodic interest rate and number of periods.**

Since the interest is compounded quarterly, the periodic interest rate is  
$$
r = \frac{8\%}{4} = 0.02.
$$

There are 4 quarters per year and the deposits are made for 6 years, so the number of deposits is  
$$
n = 4 \times 6 = 24.
$$

**Step 2. Write the future value formula for an annuity due.**

For an annuity due, the future value is given by  
$$
FV = R \left(\frac{(1+r)^n - 1}{r}\right)(1+r),
$$
where  
$$
R = 7000, \quad r = 0.02, \quad n = 24.
$$

**Step 3. Substitute the values.**

Substitute the given values into the formula:
$$
FV = 7000 \left(\frac{(1.02)^{24} - 1}{0.02}\right)(1.02).
$$

**Step 4. Compute $$ (1.02)^{24} $$.**

We use an approximation:  
$$
(1.02)^{24} \approx 1.608.
$$

Thus,  
$$
(1.02)^{24} - 1 \approx 1.608 - 1 = 0.608.
$$

Now, compute the fraction:
$$
\frac{0.608}{0.02} = 30.4.
$$

Then multiply by $$ 1.02 $$:
$$
30.4 \times 1.02 \approx 31.008.
$$

**Step 5. Multiply by the deposit amount $$ R $$.**

Now compute the future value:
$$
FV \approx 7000 \times 31.008 \approx 217056.
$$

Thus, the amount in the account at the end of 6 years is approximately $$\$217{,}056$$.

**Step 6. Compute the total amount of interest earned.**

The total amount deposited over the 24 periods is:
$$
\text{Total Deposits} = 24 \times 7000 = \$168{,}000.
$$

The interest earned is the future value minus the total deposits:
$$
\text{Interest Earned} = 217056 - 168000 = \$49056.
$$

**Final Answers:**

- The amount in the account at the end of 6 years is approximately $$\$217{,}056$$.  
- The total interest earned is approximately $$\$49{,}056$$.
At the end of 6 years, there will be approximately \$217,056 in the account. The total interest earned is approximately \$49,056.

Solución de inteligencia artificial de Upstudy

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Economics Questions