To determine the point at which the basketball reaches its maximum height, we need to analyze the given quadratic equation. The equation modeling the path of the basketball is:
[tex]\[ y = -5x^2 + 25x + 2 \][/tex]
A quadratic equation of the form [tex]\( y = ax^2 + bx + c \)[/tex] will have its maximum or minimum value at the vertex. For a parabola that opens downwards (which occurs when [tex]\( a < 0 \)[/tex]), the vertex represents the maximum point.
The x-coordinate of the vertex of a quadratic equation [tex]\( y = ax^2 + bx + c \)[/tex] can be found using the formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Here, [tex]\( a = -5 \)[/tex], [tex]\( b = 25 \)[/tex], and [tex]\( c = 2 \)[/tex]. Plugging in these values, we get:
[tex]\[ x = -\frac{25}{2(-5)} \][/tex]
[tex]\[ x = -\frac{25}{-10} \][/tex]
[tex]\[ x = 2.5 \][/tex]
This means that the basketball reaches its maximum height when [tex]\( x = 2.5 \)[/tex].
Now, we can substitute [tex]\( x = 2.5 \)[/tex] back into the quadratic equation to find the corresponding height, but since the question asks only for the x-coordinate, we already have our answer:
The basketball reaches its maximum height at [tex]\( x = 2.5 \)[/tex].
