To find the maximum gas mileage from the function \( m(x) = -0.03x^2 + 2.673x - 35.012 \), we need to determine the vertex of this quadratic function, since it opens downward (the coefficient of \( x^2 \) is negative).
The vertex \( x \) for a quadratic function in the form \( ax^2 + bx + c \) can be found using the formula:
\[ x = -\frac{b}{2a} \]
In this function:
- \( a = -0.03 \)
- \( b = 2.673 \)
Now, let's calculate \( x \):
\[ x = -\frac{2.673}{2 \cdot -0.03} = -\frac{2.673}{-0.06} = \frac{2.673}{0.06} \]
Calculating further:
\[ x \approx 44.55 \]
Now, we'll substitute this value of \( x \) back into the gas mileage function to find the maximum gas mileage:
\[ m(44.55) = -0.03(44.55^2) + 2.673(44.55) - 35.012 \]
Calculating \( 44.55^2 \):
\[ 44.55^2 \approx 1986.9025 \]
Now, substituting this value back into \( m(x) \):
\[ m(44.55) \approx -0.03(1986.9025) + 2.673(44.55) - 35.012 \] \[ \approx -59.607075 + 118.99115 - 35.012 \] \[ \approx -59.607075 + 118.99115 - 35.012 \] \[ \approx 24.372075 \]
Rounding this to one decimal place, the maximum gas mileage is:
\[ \boxed{24.4} \text{ mpg} \]
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