Question

if the rate of inflation is 2.2% 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) = 600(1.022)^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: $□

Explanation:

Step1: Find current price (t=0)

To find the current price, we substitute \( t = 0 \) into the function \( p(t)=600(1.022)^{t} \). Since any non - zero number to the power of 0 is 1, we have \( p(0)=600\times(1.022)^{0}=600\times1 = 600 \).

Step2: Find price at t = 8

To find the price 8 years from today, we substitute \( t = 8 \) into the function \( p(t)=600(1.022)^{t} \). So we calculate \( p(8)=600\times(1.022)^{8} \). First, calculate \( (1.022)^{8}\approx1.022^{8}\approx1.1916 \) (using a calculator to find the value of \( 1.022 \) raised to the 8th power). Then, multiply by 600: \( 600\times1.1916 = 714.96\approx715 \).

Answer:

Current price: $\$600$
Price 8 years from today: $\$715$