To determine the height of the highest point of the flyover, we need to find the vertex of the quadratic function [tex]\(h(x) = -3x^2 + 36x - 96\)[/tex].
A quadratic function of the form [tex]\(ax^2 + bx + c\)[/tex] reaches its maximum or minimum value at the vertex. If the coefficient [tex]\(a\)[/tex] is negative, the parabola opens downwards, which means the vertex represents the maximum point of the function.
The x-coordinate of the vertex for a quadratic function [tex]\(ax^2 + bx + c\)[/tex] is given by the formula:
[tex]\[ x_{\text{vertex}} = -\frac{b}{2a} \][/tex]
For the given function [tex]\(h(x) = -3x^2 + 36x - 96\)[/tex], we identify [tex]\(a = -3\)[/tex] and [tex]\(b = 36\)[/tex]. Substituting these values into the vertex formula, we get:
[tex]\[ x_{\text{vertex}} = -\frac{36}{2(-3)} = -\frac{36}{-6} = 6 \][/tex]
This x-coordinate, [tex]\(x_{\text{vertex}} = 6\)[/tex], corresponds to the distance at which the highest point occurs. To find the height of this highest point, we substitute [tex]\(x = 6\)[/tex] back into the original quadratic function and solve for [tex]\(h(6)\)[/tex]:
[tex]\[ h(6) = -3(6)^2 + 36(6) - 96 \][/tex]
Calculating inside the parentheses first:
[tex]\[ 6^2 = 36 \][/tex]
Now substituting back into the equation:
[tex]\[ h(6) = -3(36) + 36(6) - 96 \][/tex]
[tex]\[ h(6) = -108 + 216 - 96 \][/tex]
Combining these terms:
[tex]\[ h(6) = 108 - 96 \][/tex]
[tex]\[ h(6) = 12 \][/tex]
Therefore, the height of the highest point of the flyover is 12 feet.
