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) = 2000(1.026)^t )

find the current price of the item and the price 10 years from today.

round your answers to the nearest dollar as necessary.

Explanation:

Step1: Find current price (t=0)

Substitute \( t = 0 \) into \( p(t)=2000(1.026)^t \).
\( p(0)=2000(1.026)^0 = 2000\times1 = 2000 \)

Step2: Find price at t=10

Substitute \( t = 10 \) into \( p(t)=2000(1.026)^t \).
\( p(10)=2000(1.026)^{10} \)
Calculate \( (1.026)^{10} \approx 1.2938 \)
Then \( p(10) \approx 2000\times1.2938 = 2587.6 \approx 2588 \)

Answer:

Current price: \$2000
Price in 10 years: \$2588