Evaluating an exponential function that models a real-world situation If the rate of inflation is \( 2.5 \% \) per year, the future price \( p(t) \) (in dollars) of a certain item can be modeled by the following exponential function, where \( t \) is number of years from today. Find the price of the item 4 years from today and 8 years from today. Round your answers to the nearest dollar as necessary. Price 4 years from today: \( \$ \square \) Price 8 years from today: \( \$ \square \)

To evaluate the future price of an item based on an annual inflation rate, you'll use the exponential growth formula: \[ p(t) = p_0 \times (1 + r)^t \] Where: - \( p(t) \) = future price after \( t \) years - \( p_0 \) = current price of the item - \( r \) = annual inflation rate (expressed as a decimal) - \( t \) = number of years from today Given: - **Annual Inflation Rate, \( r \)**: \( 2.5\% \) or \( 0.025 \) in decimal form **Formula Specific to This Problem:** \[ p(t) = p_0 \times (1.025)^t \] ### Step-by-Step Calculation 1. **Identify the Current Price (\( p_0 \))**: *Note*: The current price (\( p_0 \)) of the item is essential to perform numerical calculations. Since it's not provided in your question, we'll proceed with the formula. If you have the current price, you can substitute it directly into the formula. 2. **Calculate Price After 4 Years (\( t = 4 \))**: \[ p(4) = p_0 \times (1.025)^4 \] \[ p(4) = p_0 \times 1.103812890625 \] \[ p(4) \approx p_0 \times 1.104 \] *This means the price will be approximately 10.4% higher after 4 years.* 3. **Calculate Price After 8 Years (\( t = 8 \))**: \[ p(8) = p_0 \times (1.025)^8 \] \[ p(8) = p_0 \times 1.218402 \] \[ p(8) \approx p_0 \times 1.218 \] *This means the price will be approximately 21.8% higher after 8 years.* ### Example with a Current Price If, for example, the current price of the item is **\$200**, here's how you'd compute the future prices: 1. **After 4 Years**: \[ p(4) = 200 \times 1.104 \approx \$221 \] 2. **After 8 Years**: \[ p(8) = 200 \times 1.218 \approx \$244 \] ### Final Answers Since the current price (\( p_0 \)) isn't provided, the future prices are expressed in terms of \( p_0 \): - **Price 4 years from today**: \( p(4) = p_0 \times 1.104 \) - **Price 8 years from today**: \( p(8) = p_0 \times 1.218 \) **If you provide the current price, I can help you calculate the exact future prices rounded to the nearest dollar.**