A company will need ​$30 comma 000 in 5 years for a new addition. To meet this​ goal, the company deposits money in an account today that pays 10​% annual interest compounded quarterly. Find the amount that should be invested to total ​$30 comma 000 in 5 years.
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Part 1
The company should invest ​$
  
enter your response here.
​(Do not round until the final answer. Then round to the nearest dollar as​ needed.)

To find the initial amount (principal) that the company needs to deposit today to accumulate $30,000 in 5 years with 10% annual interest compounded quarterly, we can use the compound interest formula:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

Where:
- $ A $ = future amount (total after interest), which is $30,000,
- $ P $ = initial principal (the amount we need to find),
- $ r $ = annual interest rate, which is 10% or 0.10,
- $ n $ = number of times interest is compounded per year, which is quarterly (4 times per year),
- $ t $ = time in years, which is 5 years.

### Step 1: Set up the equation
Plugging in the values we know:

\[
30000 = P \left(1 + \frac{0.10}{4}\right)^{4 \times 5}
\]

### Step 2: Simplify the expression
1. Calculate the quarterly interest rate: $ \frac{0.10}{4} = 0.025 $.
2. Calculate the total number of compounding periods: $ 4 \times 5 = 20 $.

So the equation becomes:

\[
30000 = P \left(1 + 0.025\right)^{20}
\]

### Step 3: Calculate $ (1 + 0.025)^{20} $
\[
(1 + 0.025)^{20} \approx 1.63862
\]

### Step 4: Solve for $ P $
Now, divide both sides by 1.63862 to isolate $ P $:

\[
P = \frac{30000}{1.63862} \approx 18312.20
\]

### Final Answer
The company should invest approximately **$18,312** today to reach $30,000 in 5 years with 10% interest compounded quarterly.

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Would you like more details on any step or have questions?

Here are some related questions to explore:
1. What if the interest were compounded monthly instead of quarterly?
2. How would the required investment change if the interest rate were different?
3. What if the investment period were extended to 10 years?
4. How much would the investment grow if compounded continuously?
5. How does quarterly compounding compare to simple interest over the same period?

**Tip:** When calculating compound interest, breaking down each component of the formula step-by-step can help avoid errors and clarify the process.

A company will need $30,000 in 5 years for a new addition. To meet this goal, the company deposits money in an account today that pays 10% annual interest compounded quarterly. Find the amount that should be invested to total $30,000 in 5 years.

Determine the initial investment needed for a $30,000 goal in 5 years with a 10% annual interest rate compounded quarterly. This problem covers compound interest calculations, exponential growth, and financial planning, making it suitable for Grades 10-12.

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