The problem provided involves an exponential function modeling the future price of an item given an inflation rate of 2.8% per year. The function for the price, $p(t)$, in terms of years from today $t$, is:

$p(t) = 600(1.028)^t$

We are tasked with finding two things:

1. The current price of the item (which corresponds to $t = 0$).
2. The price of the item 9 years from today (which corresponds to $t = 9$).

1. Current price:

At $t = 0$: $p(0) = 600(1.028)^0 = 600 \times 1 = 600$ Thus, the current price of the item is $600.

2. Price 9 years from today:

At $t = 9$: $p(9) = 600(1.028)^9$ Now calculating this: $p(9) \approx 600 \times (1.028)^9 \approx 600 \times 1.2682 = 760.92$ Rounding to the nearest dollar, the price 9 years from today is $761.

Final Answers:

• Current price: $600
• Price 9 years from today: $761

Would you like further details on this calculation, or have any other questions?

Here are five related questions for further exploration:

1. How does increasing the inflation rate affect the future price?
2. What would the price be 5 years from today using the same model?
3. How would the model change if the inflation rate were 3% instead of 2.8%?
4. How does compound interest differ from simple interest in this context?
5. What is the mathematical meaning of exponential growth in financial terms?

Tip: To estimate future prices using inflation, understanding exponential growth is key because prices increase at an accelerating rate with time.