2300 dollars is placed in an account with an annual interest rate of 5.5%. To the nearest tenth of a year, how long will it take for the account value to reach 4300 dollars?

To solve this problem, we use the formula for compound interest when no additional contributions are made:

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

Where:
- $ A $ is the amount of money after time $ t $ (which is \$4300 in this case),
- $ P $ is the principal amount (the initial deposit, which is \$2300),
- $ r $ is the annual interest rate (5.5% or 0.055 as a decimal),
- $ t $ is the time in years,
- $ A = 4300 $ and $ P = 2300 $.

We want to find $ t $, the time it will take for the account to grow from \$2300 to \$4300. We plug the values into the equation:

\[
4300 = 2300(1 + 0.055)^t
\]

Now, solve for $ t $.

### Step 1: Divide both sides by 2300
\[
\frac{4300}{2300} = (1.055)^t
\]

\[
1.8696 = (1.055)^t
\]

### Step 2: Take the natural logarithm of both sides
\[
\ln(1.8696) = \ln((1.055)^t)
\]

\[
\ln(1.8696) = t \cdot \ln(1.055)
\]

### Step 3: Solve for $ t $
\[
t = \frac{\ln(1.8696)}{\ln(1.055)}
\]

Now, calculating this:

\[
t \approx \frac{0.6251}{0.0535} \approx 11.7 \text{ years}
\]

Therefore, it will take approximately **11.7 years** for the account to grow from \$2300 to \$4300.

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

### Relative Questions:
1. What happens if the interest rate increases to 6%?
2. How does the formula change if the interest is compounded monthly instead of annually?
3. What is the effect of doubling the initial deposit on the time taken to reach \$4300?
4. How long would it take for the account to double at the same interest rate?
5. What would be the account value after 20 years at this interest rate?

### Tip:
Remember, increasing the frequency of compounding (e.g., monthly or daily) will result in the money growing faster than with annual compounding.

Discover how to calculate the time required for $2300 to grow to $4300 with a 5.5% annual interest rate. This detailed solution covers key mathematical concepts such as compound interest, logarithmic functions, and exponential growth, ideal for high school students in Grades 10-12.

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