Elijah invested $280 in an account paying an interest rate of 2.1% compounded continuously. Assuming no deposits or withdrawals are made, how long would it take, *to the nearest tenth of a year*, for the value of the account to reach $360?

To solve this, we will use the formula for continuous compounding:

\[
A = P e^{rt}
\]

Where:
- $ A $ is the final amount in the account,
- $ P $ is the initial principal (starting amount),
- $ r $ is the interest rate (in decimal form),
- $ t $ is the time (in years),
- $ e $ is Euler's number (approximately 2.718).

### Given:
- $ A = 360 $
- $ P = 280 $
- $ r = 0.021 $ (since 2.1% = 0.021)
- We need to solve for $ t $.

### Step 1: Set up the equation
\[
360 = 280 e^{0.021t}
\]

### Step 2: Divide both sides by 280 to isolate the exponential term
\[
\frac{360}{280} = e^{0.021t}
\]

\[
1.2857 = e^{0.021t}
\]

### Step 3: Take the natural logarithm (ln) of both sides to get rid of the exponential
\[
\ln(1.2857) = 0.021t
\]

### Step 4: Solve for $ t $
\[
t = \frac{\ln(1.2857)}{0.021}
\]

Let's compute that value.

The time it would take for the value of the account to reach $360 is approximately $ 12.0 $ years, rounded to the nearest tenth.

Would you like more details or have any questions about the process?

Here are 5 related questions for further exploration:
1. How would the time change if the interest rate were 3%?
2. What if the principal was $500 instead of $280?
3. Can you derive the same result using a different compounding frequency, such as quarterly compounding?
4. How would withdrawals or additional deposits affect the time required?
5. What if the final value needed to be $500 instead of $360?

**Tip:** Continuous compounding yields the highest returns because interest is calculated and added to the account balance an infinite number of times per period.

Elijah invested $280 in an account paying an interest rate of 2.1% compounded continuously. Assuming no deposits or withdrawals are made, how long would it take, to the nearest tenth of a year, for the value of the account to reach $360?

Learn how to calculate the time required for an investment of $280 to grow to $360 with continuous compounding at a 2.1% interest rate. This problem involves exponential growth, logarithms, and continuous compounding formulas, suitable for high school students in Grades 10-12.

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