Problem

If the rate of inflation is $3.7 \%$ 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.
\[
p(t)=800(1.037)^{t}
\]
Find the current price of the item and the price 8 years from today. Round your answers to the nearest dollar as necessary.

Answer

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Answer

Final Answer: The current price of the item is \(\boxed{800}\) dollars and the price 8 years from today will be \(\boxed{1070}\) dollars.

Steps

Step 1 :The rate of inflation is given as $3.7 \%$ 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: \[p(t)=800(1.037)^{t}\]

Step 2 :To find the current price of the item, we substitute $t=0$ into the function: \[p(0)=800(1.037)^{0}\]

Step 3 :Solving the above equation gives us the current price of the item as \(\boxed{800}\) dollars.

Step 4 :To find the price of the item 8 years from now, we substitute $t=8$ into the function: \[p(8)=800(1.037)^{8}\]

Step 5 :Solving the above equation gives us the price of the item 8 years from now as \(\boxed{1070}\) dollars.

Step 6 :Final Answer: The current price of the item is \(\boxed{800}\) dollars and the price 8 years from today will be \(\boxed{1070}\) dollars.

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