If the rate of inflation is \( 3.9 \% \) per year, the future price \( p(t) \) (in dollars) of a certain item can be modeled by the following exponential function, where \( t \) is the number of years from today. \[ p(t)=1200(1.039)^{t} \] Find the current price of the item and the price 8 years from today. Round your answers to the nearest dollar as necessary. Current price: Price 8 years from today:
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The current price of the item is given directly in the function as \( p(0) = 1200(1.039)^0 = 1200(1) = 1200 \) dollars. To find the price 8 years from today, we plug in \( t = 8 \) into the function: \[ p(8) = 1200(1.039)^8 \] Calculating \( (1.039)^8 \) gives approximately \( 1.348850 \), so: \[ p(8) \approx 1200 \times 1.348850 \approx 1618.62 \] Rounding to the nearest dollar, we have \( p(8) \approx 1619 \) dollars. Therefore, the current price is \( 1200 \) dollars, and the price 8 years from today is \( 1619 \) dollars.
