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) = 800(1.022)^t find the current price of the item and the price 9 years from today. round your answers to the nearest dollar as necessary. current price: $ price 9 years from today: $

Explanation:

Step1: Find current price (t=0)

To find the current price, we substitute \( t = 0 \) into the function \( P(t)=800(1.022)^t \).
Using the property of exponents \( a^0 = 1 \) for any non - zero \( a \), we have:
\( P(0)=800(1.022)^0 = 800\times1=800 \)

Step2: Find price 9 years from today (t = 9)

We substitute \( t = 9 \) into the function \( P(t)=800(1.022)^t \).
So \( P(9)=800\times(1.022)^9 \)
First, calculate \( (1.022)^9\approx1.022^{9}\approx1.217 \) (using a calculator to find the value of the exponential)
Then \( P(9)=800\times1.217 = 973.6\approx974 \) (rounded to the nearest dollar)

Answer:

Current price: \(\$800\)
Price 9 years from today: \(\$974\)