Questions: If the rate of inflation is 3.4% 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)=800(1.034)^t] Find the current price of the item and the price 10 years from today. Round your answers to the nearest dollar as necessary. Current price: () Price 10 years from today:

Solution Steps

The current price corresponds to \( t = 0 \). Substitute \( t = 0 \) into the equation \( p(t) = 800(1.034)^t \): \[ p(0) = 800(1.034)^0 \] Since any number raised to the power of 0 is 1: \[ p(0) = 800 \times 1 = 800 \] The current price is \( \$800 \).

To find the price 10 years from today, substitute \( t = 10 \) into the equation: \[ p(10) = 800(1.034)^{10} \] Calculate \( (1.034)^{10} \): \[ (1.034)^{10} \approx 1.397 \] Multiply by 800: \[ p(10) = 800 \times 1.397 \approx 1117.6 \] Round to the nearest dollar: \[ p(10) \approx 1118 \] The price 10 years from today is \( \$1118 \).

• Current price: \( \$800 \)
• Price 10 years from today: \( \$1118 \)

Final Answer