If the rate of inflation is $$ 2.8 \% $$ per year, the future price $$ p(t) $$ (in dollars) of a certain item can be modeled by $$ t $$ exponential function, where $$ t $$ is the number of years from today. $$ p(t)=3000(1.028)^{t} $$ Find the current price of the item and the price 8 years from today. Round your answers to the nearest dollar as necessary. Current price: 8 years from today: $$ \$ \square $$

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UpStudy AI · Accepted Answer

**Step 1:**  
The current price is given by evaluating the function at $$ t=0 $$:  
$$
p(0) = 3000(1.028)^0 = 3000(1) = 3000.
$$

**Step 2:**  
The price 8 years from today is given by:  
$$
p(8) = 3000(1.028)^8.
$$

**Step 3:**  
Calculate $$ (1.028)^8 $$. Using the property of logarithms, we can compute:  
$$
(1.028)^8 = e^{8\ln(1.028)}.
$$  
Approximating, note that:  
$$
\ln(1.028) \approx 0.02763,
$$  
$$
8\ln(1.028) \approx 0.22104,
$$  
$$
e^{0.22104} \approx 1.2477.
$$

**Step 4:**  
Thus, the price 8 years from today is:  
$$
p(8) \approx 3000 \times 1.2477 \approx 3743.1.
$$  
Rounded to the nearest dollar, the price is approximately $$ \$3743 $$.

**Final Answers:**  
- Current price: $$ \$3000 $$  
- 8 years from today: $$ \$3743 $$
Current price: \$3000 8 years from today: \$3743

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