Sure, let's go through the solution step-by-step.
### Step 1: Understanding the Formula
Given the formula for the future price, which is:
[tex]\[ p(t) = 1200 \times (1.034)^t \][/tex]
Here:
- [tex]\(1200\)[/tex] is the initial price of the item.
- [tex]\(1.034\)[/tex] is the factor accounting for the annual inflation rate of [tex]\(3.4\%\)[/tex].
- [tex]\(t\)[/tex] is the number of years from today.
### Step 2: Current Price of the Item
The current price of the item is essentially the price at [tex]\(t = 0\)[/tex]:
[tex]\[ p(0) = 1200 \times (1.034)^0 \][/tex]
Since any number raised to the power of zero is one, this simplifies to:
[tex]\[ p(0) = 1200 \times 1 = 1200 \][/tex]
Thus, the current price of the item is [tex]\(\$1200\)[/tex].
### Step 3: Price 9 Years from Today
Now, we need to find the price of the item 9 years from today:
[tex]\[ p(9) = 1200 \times (1.034)^9 \][/tex]
Performing the exponentiation and multiplication (assuming the calculations are done and given as per the result):
[tex]\[ p(9) \approx 1621 \][/tex]
So, the price 9 years from today is approximately [tex]\(\$1621\)[/tex].
### Step 4: Rounding to the Nearest Dollar
Both prices are already in the whole number form, so no further rounding is needed.
### Conclusion
- The current price of the item is [tex]\(\$1200\)[/tex].
- The price 9 years from today is [tex]\(\$1621\)[/tex].
