Use the formula 

Aequals=Upper P left parenthesis 1 plus StartFraction r Over n EndFraction right parenthesis Superscript ntP1+rnnt 

to solve the compound interest problem.

Find how long it takes for 

​$20002000 

to double if it is invested at 

99​% 

interest compounded monthly.

Question content area bottom

Part 1

The money will double in value in approximately 

enter your response here 

years. 

​(Do not round until the final answer. Then round to the nearest tenth as​ needed.)

To find out how long it takes for $2000 to double at a 9% annual interest rate compounded monthly, we will use the compound interest formula:

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

Where:
- $A$ is the amount of money accumulated after n years, including interest.
- $P$ is the principal amount (initial investment).
- $r$ is the annual interest rate (decimal).
- $n$ is the number of times that interest is compounded per year.
- $t$ is the number of years the money is invested or borrowed.

### Given:
- $P = 2000$
- $A = 4000$ (since we want it to double)
- $r = 0.09$ (9% as a decimal)
- $n = 12$ (monthly compounding)

### Step 1: Substitute the values into the formula

\[
4000 = 2000 \left(1 + \frac{0.09}{12}\right)^{12t}
\]

### Step 2: Simplify the equation

\[
2 = \left(1 + \frac{0.09}{12}\right)^{12t}
\]

Calculate $1 + \frac{0.09}{12}$:

\[
1 + \frac{0.09}{12} = 1 + 0.0075 = 1.0075
\]

So the equation now is:

\[
2 = (1.0075)^{12t}
\]

### Step 3: Take the logarithm of both sides

Taking the natural logarithm:

\[
\ln(2) = \ln((1.0075)^{12t})
\]

Using the power rule of logarithms:

\[
\ln(2) = 12t \cdot \ln(1.0075)
\]

### Step 4: Solve for $t$

\[
t = \frac{\ln(2)}{12 \cdot \ln(1.0075)}
\]

### Step 5: Calculate the values

Calculate $\ln(2)$ and $\ln(1.0075)$:

\[
\ln(2) \approx 0.693147
\]
\[
\ln(1.0075) \approx 0.007471
\]

Substituting back into the equation:

\[
t \approx \frac{0.693147}{12 \cdot 0.007471} \approx \frac{0.693147}{0.089652} \approx 7.74
\]

### Final Answer

The money will double in value in approximately **7.7 years** (rounded to the nearest tenth).

### Questions for Further Exploration:
1. How does changing the interest rate affect the time to double the investment?
2. What are other methods to calculate the time needed to double an investment?
3. How would this calculation differ for daily compounding instead of monthly?
4. What are the risks associated with investing in high-interest accounts?
5. Can you explain the concept of the "Rule of 72" for estimating doubling time?

### Tip
Always check your calculations by plugging your result back into the original formula to ensure consistency.

Find how long it takes for $2000 to double if it is invested at 9% interest compounded monthly.

Discover how to calculate the time required for $2000 to double at a 9% annual interest rate compounded monthly. This guide includes a detailed solution using the compound interest formula and logarithms, ideal for high school students.

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