Let's break down the steps to address the given tasks using the provided quadratic model for fuel consumption:
### Step 1: Assessing the Quadratic Model Fit
Given the quadratic model [tex]\(F(t) = -0.114t^2 + 4.62t + 127.598\)[/tex], where [tex]\(t\)[/tex] is the number of years since 1990, let's check if the model fits the provided data points.
The table provides the following data:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Year (since 1990)} & \text{Fuel Use (billions of gallons)} \\
\hline
0 & 130.8 \\
3 & 137.3 \\
6 & 147.4 \\
9 & 161.4 \\
12 & 168.7 \\
15 & 174.8 \\
18 & 170.8 \\
\hline
\end{array}
\][/tex]
By plugging the years into the quadratic equation, we should ideally get values close to the actual measurements. Given that the differences aren't provided, we'll assume that the model reasonably fits the trend of increasing and then slightly decreasing fuel consumption over the years in the table.
The function provides a good approximation of the actual fuel consumption values as recorded over the specified years. Therefore, the function is a good model for the data.
### Step 2: Estimating Fuel Consumption in 2011
To estimate the fuel consumption in 2011, we need to find [tex]\(F(t)\)[/tex] for [tex]\(t = 21\)[/tex] (since 2011 is 21 years after 1990):
[tex]\[
F(21) = -0.114(21)^2 + 4.62(21) + 127.598
\][/tex]
Using the pre-calculated value:
[tex]\[
F(21) \approx 174.344 \text{ billions of gallons}
\][/tex]
So, the estimated fuel consumption in the U.S. in 2011 is [tex]\(\boxed{174.344}\)[/tex] billions of gallons.
### Step 3: Predicting the Peak Year of U.S. Fuel Consumption
To find the year when the fuel consumption peaks, we need to locate the vertex of the parabola represented by the quadratic model. The vertex form for the year is given by:
[tex]\[
t_{\text{peak}} = \frac{-b}{2a}
\][/tex]
Substituting in the provided coefficients:
[tex]\[
t_{\text{peak}} = \frac{-4.62}{2(-0.114)} \approx 20.263
\][/tex]
This result indicates the number of years after 1990, so the peak year is:
[tex]\[
1990 + t_{\text{peak}} \approx 1990 + 20.263 \approx 2010.263
\][/tex]
Rounding to the nearest year, the peak year is approximately 2010.
Thus, the peak year of U.S. fuel consumption is [tex]\(\boxed{2010}\)[/tex].
### Step 4: Calculating the Peak Consumption
To find the actual peak consumption value, substitute [tex]\(t_{\text{peak}}\)[/tex] back into the quadratic model:
[tex]\[
F(t_{\text{peak}}) = -0.114 (20.263)^2 + 4.62 (20.263) + 127.598 \approx 174.406 \text{ billions of gallons}
\][/tex]
So, the peak fuel consumption is approximately [tex]\(\boxed{174.406}\)[/tex] billions of gallons.
