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:

To find the current price of the item and the price 8 years from today, we need to calculate the values of \( p(0) \) and \( p(8) \) using the given exponential function \( p(t) = 1200(1.039)^{t} \). 1. Current price: To find the current price, we substitute \( t = 0 \) into the function: \[ p(0) = 1200(1.039)^{0} \] Since any number raised to the power of 0 is 1, the current price is: \[ p(0) = 1200 \] 2. Price 8 years from today: To find the price 8 years from today, we substitute \( t = 8 \) into the function: \[ p(8) = 1200(1.039)^{8} \] We can calculate this value using a calculator or by simplifying the expression: \[ p(8) = 1200(1.039)^{8} \] Now, let's calculate the values of \( p(0) \) and \( p(8) \) to find the current price and the price 8 years from today. Calculate the value by following steps: - step0: Calculate: \(1200\times 1.039^{0}\) - step1: Evaluate the power: \(1200\times 1\) - step2: Multiply: \(1200\) Calculate or simplify the expression \( 1200*(1.039)^8 \). Calculate the value by following steps: - step0: Calculate: \(1200\times 1.039^{8}\) - step1: Convert the expressions: \(1200\left(\frac{1039}{1000}\right)^{8}\) - step2: Simplify: \(1200\times \frac{1039^{8}}{1000^{8}}\) - step3: Rewrite the expression: \(200\times 6\times \frac{1039^{8}}{1000^{8}}\) - step4: Rewrite the expression: \(200\times 6\times \frac{1039^{8}}{200^{8}\times 5^{8}}\) - step5: Reduce the numbers: \(6\times \frac{1039^{8}}{200^{7}\times 5^{8}}\) - step6: Rewrite the expression: \(2\times 3\times \frac{1039^{8}}{200^{7}\times 5^{8}}\) - step7: Rewrite the expression: \(2\times 3\times \frac{1039^{8}}{8^{7}\times 25^{7}\times 5^{8}}\) - step8: Rewrite the expression: \(2\times 3\times \frac{1039^{8}}{2^{21}\times 25^{7}\times 5^{8}}\) - step9: Reduce the numbers: \(3\times \frac{1039^{8}}{2^{20}\times 25^{7}\times 5^{8}}\) - step10: Multiply the fractions: \(\frac{3\times 1039^{8}}{2^{20}\times 5^{22}}\) The current price of the item is $1200, and the price 8 years from today is approximately $1629.69.