Question

if the rate of inflation is 4.1% 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.

$p(t) = 600(1.041)^t$

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: $\square$

price 8 years from today: $\square$

Explanation:

Step1: Find current price (t=0)

To find the current price, we substitute \( t = 0 \) into the function \( p(t)=600(1.041)^t \). Since any non - zero number to the power of 0 is 1, we have \( p(0)=600\times(1.041)^0 = 600\times1=600 \).

Step2: Find price at t = 8

We substitute \( t = 8 \) into the function \( p(t)=600(1.041)^t \). So we calculate \( p(8)=600\times(1.041)^8 \). First, calculate \( (1.041)^8\approx1.041\times1.041\times1.041\times1.041\times1.041\times1.041\times1.041\times1.041\approx1.377 \) (using a calculator for more accurate calculation, \( (1.041)^8\approx e^{8\ln(1.041)}\approx e^{8\times0.0398}\approx e^{0.3184}\approx1.375 \)). Then \( p(8)=600\times1.375 = 825 \) (more accurately, using a calculator to compute \( 600\times(1.041)^8\): \( (1.041)^8\approx1.377127 \), and \( 600\times1.377127\approx826.2762\approx826 \)).

Answer:

Current price: $\$600$
Price 8 years from today: $\$826$