Bot GPT 4-o mini 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:
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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.
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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!
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