Question

if the rate of inflation is 2.6% 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) = 400(1.026)^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)

Substitute \( t = 0 \) into \( p(t)=400(1.026)^t \).
\( p(0)=400(1.026)^0 = 400\times1 = 400 \)

Step2: Find price at t=9

Substitute \( t = 9 \) into \( p(t)=400(1.026)^t \).
\( p(9)=400(1.026)^9 \)
Calculate \( (1.026)^9 \approx 1.2568 \) (using a calculator)
Then \( p(9) \approx 400\times1.2568 = 502.72 \approx 503 \)

Answer:

Current price: $\$400$
Price 9 years from today: $\$503$