Question

Andrea invested $\$3,000$ in an online currency. If the value of the currency increases at a rate of 14\% per year, the situation can be modeled using the equation A(t) = 3,000(1.14)^t, where A is the amount in the investment after t years.

Answer 1

It looks like you have a situation involving an investment with exponential growth. The equation you've provided, \( A(t) = 3000(1.14)^t \), is indeed the correct model for your investment. Here’s a detailed breakdown of this model:

- \( A(t) \) is the amount of money in the investment after \( t \) years.
- The initial investment amount (also called the principal) is \$3,000.
- The growth rate is 14% per year, which is expressed as 1.14 when calculating the total value (1 + 0.14).

### To Explore This Further:

1. **Calculating Future Values:**
You can calculate the amount in the investment after a certain number of years by substituting the value of \( t \) into the equation. For example, to find out how much the investment will be worth after 5 years:

\[
A(5) = 3000(1.14)^5
\]

This can be computed as follows:

\[
A(5) \approx 3000 \times (1.9254) \approx 5776.20
\]

Hence, after 5 years, the investment would be approximately \$5,776.20.

2. **Finding Time for a Certain Amount:**
If you want to determine how long it will take for the investment to reach a specific amount, say \$5,000, you would set the equation equal to that amount and solve for \( t \):

\[
5000 = 3000(1.14)^t
\]

Dividing by 3,000 gives:

\[
\frac{5000}{3000} = (1.14)^t
\]

\[
\frac{5}{3} \approx 1.6667 = (1.14)^t
\]

To solve for \( t \), take the logarithm of both sides:

\[
\log(1.6667) = t \cdot \log(1.14)
\]

Then, isolate \( t \):

\[
t = \frac{\log(1.6667)}{\log(1.14)}
\]

Calculating this will give you the time in years needed to reach \$5,000.

### Summary
The investment grows exponentially due to the compounding effect of the interest rate. The formula \( A(t) = 3000(1.14)^t \) allows you to calculate the future value of the investment or to determine other variables such as the time required to reach a specific investment goal. If you have any specific queries regarding calculations or another aspect of this investment model, feel free to ask!