To find the maximum height of the ball modeled by the function \( h(x) = -(x+1)(x-7) \), we first need to rewrite the function in vertex form, which makes it easier to identify the maximum or minimum value.
We can start by expanding the given function:
\[ h(x) = -(x+1)(x-7) \]
Expanding the expression:
\[ h(x) = -((x)(x) + (7)(x) - (1)(x) - (7)) \] \[ = - (x^2 + 6x - 7) \] \[ = -x^2 - 6x + 7 \]
Now, we will rewrite this quadratic equation in the vertex form \(a(x - h)^2 + k\) by completing the square. The quadratic function has the form:
\[ h(x) = -1(x^2 + 6x) + 7 \]
To complete the square on \(x^2 + 6x\):
- Take half of the coefficient of \(x\) (which is \(6\)), square it \((\frac{6}{2})^2 = 9\), and add and subtract this value inside the parentheses.
- This gives us:
\[ h(x) = -1(x^2 + 6x + 9 - 9) + 7 \] \[ = -1((x + 3)^2 - 9) + 7 \] \[ = -1(x + 3)^2 + 9 + 7 \] \[ = -1(x + 3)^2 + 16 \]
Now, we can see that the vertex of the parabola is at \((-3, 16)\), which represents the maximum height since the parabola opens downwards (indicated by the negative coefficient).
Thus, the maximum height that the ball will reach is:
\[ \boxed{16} \text{ meters} \]