Write out all the written work that would go into solving this problem. determine how much time is required  for a 500$ investment to double if the interest rate is 4.75% compounded annually

We are given that a $500 investment needs to double, and the interest rate is 4.75% compounded annually. We can use the **compound interest formula** to solve for the time required.

### Step 1: Write down the compound interest formula.

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

Where:
- $ A $ is the amount of money after interest.
- $ P $ is the principal (initial investment).
- $ 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.

In this case:
- $ A = 2P $ (since the investment is doubling).
- $ P = 500 $ (initial investment).
- $ r = 0.0475 $ (4.75% expressed as a decimal).
- $ n = 1 $ (compounded annually).

### Step 2: Substitute values into the formula.

Since $ A = 2P $, we have:

\[
2P = P \left( 1 + \frac{0.0475}{1} \right)^{1 \cdot t}
\]

Canceling $ P $ from both sides:

\[
2 = \left( 1 + 0.0475 \right)^t
\]

This simplifies to:

\[
2 = (1.0475)^t
\]

### Step 3: Solve for $ t $.

Take the natural logarithm of both sides to bring the exponent down:

\[
\ln(2) = \ln\left( (1.0475)^t \right)
\]

Using the logarithmic identity $ \ln(a^b) = b \cdot \ln(a) $:

\[
\ln(2) = t \cdot \ln(1.0475)
\]

Now, solve for $ t $:

\[
t = \frac{\ln(2)}{\ln(1.0475)}
\]

### Step 4: Compute the values.

First, calculate the natural logarithms:

\[
\ln(2) \approx 0.6931
\]
\[
\ln(1.0475) \approx 0.0464
\]

Now substitute these values into the equation for $ t $:

\[
t = \frac{0.6931}{0.0464} \approx 14.93
\]

### Step 5: Interpret the result.

It will take approximately **14.93 years** for the $500 investment to double at an interest rate of 4.75%, compounded annually.

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

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Here are 5 related questions to consider:
1. What if the interest was compounded monthly instead of annually?
2. How long would it take for the investment to triple at the same interest rate?
3. What would the formula look like if the interest rate were 6%?
4. How does changing the principal amount affect the time required to double?
5. What is the effect of continuous compounding on the time to double?

**Tip:** Always express percentages as decimals when using them in formulas (e.g., 4.75% becomes 0.0475).

Learn how long it takes for a $500 investment to double with a 4.75% annual interest rate using the compound interest formula. Detailed step-by-step solution provided.

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