For a total accumulated amount of ​$8427.85​, a principal of ​$3000​, and a time period of 8 ​years, use the compound interest formula to find​ r, if interest is compounded monthly.
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Part 1
requals
  
enter your response here​%
​(Type an integer or decimal rounded to the nearest hundredth as​ needed.)

We can use the compound interest formula to find the interest rate, $ r $:

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

Where:
- $ A $ is the total accumulated amount (8427.85)
- $ P $ is the principal amount (3000)
- $ r $ is the annual interest rate (which we need to find)
- $ n $ is the number of times the interest is compounded per year (monthly, so $ n = 12 $)
- $ t $ is the time in years (8 years)

### Step 1: Substitute the known values into the formula:
\[
8427.85 = 3000 \left( 1 + \frac{r}{12} \right)^{12 \times 8}
\]

Simplifying the exponent:
\[
8427.85 = 3000 \left( 1 + \frac{r}{12} \right)^{96}
\]

### Step 2: Solve for $ r $

We can isolate $ \left( 1 + \frac{r}{12} \right)^{96} $ and use logarithms to solve for $ r $.

I'll solve this step-by-step to find the value of $ r $.The annual interest rate $ r $ is approximately **12.98%**.

Would you like more details on the steps or have any further questions?

Here are some related questions for further exploration:
1. How would the interest rate change if compounded quarterly instead of monthly?
2. What would the total amount be after 10 years with the same rate?
3. How does simple interest compare to compound interest over the same period?
4. What if the initial principal was $5000 instead of $3000, keeping other variables the same?
5. How would the total amount change if the interest was compounded daily?

**Tip:** Compounding more frequently (monthly vs annually) leads to more interest accumulated over time.

For a total accumulated amount of $8427.85, a principal of $3000, and a time period of 8 years, use the compound interest formula to find r, if interest is compounded monthly.

Discover how to calculate the annual interest rate for a total accumulated amount of $8427.85, starting from a $3000 principal, over 8 years with monthly compounding. This solution uses the compound interest formula and logarithmic functions, ideal for high school students learning exponential growth.

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