To determine the horizontal distance at which the long jumper reaches their maximum height, we need to analyze the quadratic equation modeling the height of the jumper:
[tex]\[ h(x) = -0.05x^2 + 0.363x. \][/tex]
This equation is in the standard form of a quadratic equation:
[tex]\[ h(x) = ax^2 + bx + c, \][/tex]
where \(a = -0.05\), \(b = 0.363\), and \(c\) is implicitly zero in this context (although it is not relevant for finding the vertex).
The vertex of a parabola represented by the quadratic equation \( ax^2 + bx + c \) gives us the maximum or minimum point of the parabola. For parabolas that open downwards, like this one (since \(a < 0\)), the vertex represents the maximum point.
The x-coordinate of the vertex can be found using the formula:
[tex]\[ x = -\frac{b}{2a}. \][/tex]
Substituting the values of \(a\) and \(b\):
[tex]\[ a = -0.05, \][/tex]
[tex]\[ b = 0.363, \][/tex]
we get:
[tex]\[ x = -\frac{0.363}{2 \times -0.05}. \][/tex]
Calculating the value:
[tex]\[ x = -\frac{0.363}{-0.1} = 3.63. \][/tex]
Therefore, the horizontal distance at which the maximum height occurs is approximately:
[tex]\[ x \approx 3.63 \text{ meters}. \][/tex]
So, the long jumper reaches a maximum height when the horizontal distance from the point of launch is approximately [tex]\(\boxed{3.63}\)[/tex] meters.
