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. Identify the key features of the graph and interpret the meaning of the data

To analyze the situation modeled by the equation \( A(t) = 3000(1.14)^t \), let's identify the key features of the graph and interpret the meaning of the data.

Key Features:

1. Initial Value (A(0)):

   • At \( t = 0 \), the initial investment \( A(0) = 3000(1.14)^0 = 3000 \).
   • This means Andrea starts with an investment of $3,000.

2. Growth Rate:

   • The base of the exponent, \( 1.14 \), indicates that the investment grows by 14% each year.
   • This means that every year, the amount is multiplied by 1.14, reflecting the compounding nature of the investment.

3. Long-term Behavior:

   • As \( t \) (the number of years) increases, \( A(t) \) grows exponentially. This is characteristic of exponential growth, as the investment will increase significantly over time due to compounding.

4. Y-intercept:

   • The y-intercept of the graph is at the point (0, 3000), reflecting the initial investment amount.

5. Asymptotic Behavior:

   • As \( t \) approaches infinity, \( A(t) \) will also approach infinity. There is no upper limit to the amount since the investment can keep growing.

6. Doubling Time:

   • To find out when the investment might double, we can use the rule of 70, which approximates the doubling time as \( \frac{70}{\text{growth rate}} \).
   • In this case, \( \frac{70}{14} \approx 5 \) years for the investment to double (though this is an approximation).

Interpretation of the Data:

• Exponential Growth: The model demonstrates exponential growth of Andrea's investment due to the annual compounded growth. This means that not only is the principal increasing, but the interest earned is also generating additional interest in subsequent years.

• Financial Planning: Investors can use this model to forecast future financial goals. For example, after 5 years (\( A(5) \)), Andrea would have approximately \( 3000(1.14)^5 \approx 3000 \times 1.925 = 5775 \). This might help Andrea make decisions about her investment strategy.

• Investing in Growth: The 14% growth rate is significant and indicates that investing in this particular online currency could yield substantial returns over time. This highlights the potential benefits of investing in high-growth assets.

• Risk Consideration: While the model is optimistic about growth, it’s important to recognize that such high returns often come with increased risks. Future performance may not guarantee the same growth, and market conditions can change.

Overall, the equation reflects a promising investment opportunity for Andrea, given the high growth rate and the nature of compound interest.