A bank account has an initial deposit of $ 14 comma 000$14,000. After 88 ​years, the bank account has ​$20 comma 00020,000. Assuming that compounding interest is done annually and that no withdrawals are​ taken, find the annual interest rate on the investment.
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Question content area right
Part 1
If interest is compounded​ annually, what is the approximate interest rate on the​ investment?
enter your response here​%
​(Do not round until the final answer. Then round to the nearest​ hundredth.)

To find the annual interest rate $ r $ when interest is compounded annually, we can use the compound interest formula:

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

Where:
- $ A $ is the final amount in the account,
- $ P $ is the initial principal (deposit),
- $ r $ is the annual interest rate (as a decimal),
- $ n $ is the number of times the interest is compounded per year,
- $ t $ is the time (in years).

Since interest is compounded annually, $ n = 1 $, and we can simplify the formula to:

\[
A = P(1 + r)^t
\]

We are given:
- $ A = 20,000 $,
- $ P = 14,000 $,
- $ t = 8 $ years.

Now, plug in the values and solve for $ r $:

\[
20,000 = 14,000 (1 + r)^8
\]

### Step-by-step Solution:

1. Divide both sides by 14,000:

\[
\frac{20,000}{14,000} = (1 + r)^8
\]

\[
1.42857 = (1 + r)^8
\]

2. Take the 8th root of both sides to solve for $ 1 + r $:

\[
(1 + r) = \left( 1.42857 \right)^{\frac{1}{8}}
\]

3. Subtract 1 from both sides to find $ r $:

\[
r = \left( 1.42857 \right)^{\frac{1}{8}} - 1
\]

Let me calculate this for the precise value.The approximate annual interest rate is $ 4.56\% $, rounded to the nearest hundredth.

Would you like more details on any part of the process or another explanation?

Here are 5 related questions to explore:
1. What is the formula for compound interest if compounded quarterly?
2. How does the frequency of compounding affect the interest rate?
3. What would the final amount be with a different initial deposit but the same interest rate?
4. How do you calculate simple interest compared to compound interest?
5. How would the interest rate change if the time period was doubled?

**Tip:** Always convert percentages back to decimals when using them in equations.

A bank account has an initial deposit of $14,000. After 8 years, the bank account has $20,000. Assuming that compounding interest is done annually and that no withdrawals are taken, find the annual interest rate on the investment.

This problem involves calculating the annual interest rate for a $14,000 investment that grows to $20,000 in 8 years with annual compounding. Learn the step-by-step process using the compound interest formula and exponential growth concepts. Suitable for Grades 10-12.

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