Question

Write an exponential function for the situation below and use your equation to answer the question.

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Camilo purchased a rare coin from a dealer for $300. The value of the coin increases 5% each year. Determine the value of the coin in 5 years.

Answer 1

To model the situation with an exponential function, we can use the formula for exponential growth:

\[ V(t) = V_0 \cdot (1 + r)^t \]

Where:
- \( V(t) \) is the value of the coin after \( t \) years,
- \( V_0 \) is the initial value of the coin,
- \( r \) is the growth rate (as a decimal),
- \( t \) is the number of years.

In this case:
- \( V_0 = 300 \) (the initial value of the coin),
- \( r = 0.05 \) (5% growth rate expressed as a decimal),
- \( t = 5 \) (the number of years).

Now we can plug the values into the formula:

\[ V(5) = 300 \cdot (1 + 0.05)^5 \]

Calculating this step-by-step:

1. Calculate \( 1 + 0.05 \):
\[ 1 + 0.05 = 1.05 \]

2. Raise \( 1.05 \) to the power of \( 5 \):
\[ 1.05^5 \approx 1.2762815625 \]

3. Multiply by the initial value:
\[ V(5) \approx 300 \cdot 1.2762815625 \approx 382.88 \]

Thus, the value of the coin in 5 years will be approximately $382.88.