Problem

If the rate of inflation is $3.9 \%$ 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.
\[
p(t)=2000(1.039)^{t}
\]
Find the current price of the item and the price 10 years from today.
Round your answers to the nearest dollar as necessary.
Current price:
Price 10 years from today:

Answer

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Answer

Final Answer: The current price of the item is \(\boxed{2000}\) dollars and the price 10 years from today will be \(\boxed{2932}\) dollars.

Steps

Step 1 :The rate of inflation is given as $3.9 \%$ per year. This can be written as a decimal as 0.039.

Step 2 :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: \[p(t)=2000(1.039)^{t}\]

Step 3 :The current price of the item is given by the function $p(t)$ when $t=0$. Substituting $t=0$ into the equation gives: \[p(0)=2000(1.039)^{0}=2000\]

Step 4 :The price 10 years from today is given by the function $p(t)$ when $t=10$. Substituting $t=10$ into the equation gives: \[p(10)=2000(1.039)^{10}\]

Step 5 :Calculating the above expression gives a value of approximately 2932.

Step 6 :Final Answer: The current price of the item is \(\boxed{2000}\) dollars and the price 10 years from today will be \(\boxed{2932}\) dollars.

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