To solve the problem:

The function for gas mileage is given as:

$m(x) = -0.027x^2 + 2.659x - 35.013$

To determine the speed ($x$) at which the car gets its maximum mileage, we need to find the vertex of the quadratic function. Since this is a parabola that opens downward ($-0.027 < 0$), the maximum occurs at the vertex. The formula for the $x$-coordinate of the vertex is:

$x = -\frac{b}{2a}$

Here, $a = -0.027$ and $b = 2.659$. Substituting these values:

$x = -\frac{2.659}{2(-0.027)}$

Let me calculate that. Coefficients of the quadratic equation a = -0.027 b = 2.659

Calculating the x-coordinate of the vertex

x_vertex = -b / (2 * a) x_vertexThe speed at which the car achieves its maximum gas mileage is approximately 49 mph when rounded to the nearest whole number.

Would you like further explanation or assistance with this problem? Here are some related questions to deepen your understanding:

1. What is the maximum gas mileage achieved by the vehicle at this speed?
2. How do you determine whether a quadratic function has a maximum or minimum?
3. Can you solve the problem graphically and confirm the vertex?
4. How would the solution change if the coefficient $a$ were positive instead of negative?
5. Could there be real-world limitations to this mathematical model?

Tip: Remember that the vertex formula $-\frac{b}{2a}$ works only for quadratic equations in standard form: $ax^2 + bx + c$.