Question

If the rate of inflation is 3.7% per year, the furture 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 10 years from today.
Round your answers to the nearest dollar as necessary.
Current price: $_
Price 10 years from today: $_

Answer 1

To find the current price of the item, we can plug \( t = 0 \) into the function \( p(t) = 800(1.037)^t \).

### Current Price:
\[
p(0) = 800(1.037)^0 = 800(1) = 800
\]
So, the current price of the item is **$800**.

### Price 10 Years from Today:
Now, we will find the price 10 years from today by plugging \( t = 10 \) into the function.

\[
p(10) = 800(1.037)^{10}
\]

Calculating \( 1.037^{10} \):
\[
1.037^{10} \approx 1.4107
\]

Now, calculate \( p(10) \):
\[
p(10) \approx 800 \times 1.4107 \approx 1128.56
\]

Rounding to the nearest dollar:
\[
p(10) \approx 1129
\]

### Final Answers:
- Current price: **$800**
- Price 10 years from today: **$1129**