To find the time at which the stone reaches its maximum height, we can analyze the quadratic function given by:
\[ w(x) = -5(x - 8)(x + 4) \]
This is a quadratic equation in vertex form, \(w(x) = a(x - h)^2 + k\), where \((h, k)\) is the vertex of the parabola. Since the coefficient \(a\) is negative (-5), this means the parabola opens downwards, and therefore the vertex represents the maximum point.
To find the vertex, we first need to expand the given function:
\[ w(x) = -5[(x - 8)(x + 4)] = -5[x^2 + 4x - 8x - 32] = -5[x^2 - 4x - 32] \]
Now expanding it fully:
\[ w(x) = -5x^2 + 20x + 160 \]
The time at which the maximum height occurs can be found using the formula for the vertex of a parabola, given by \(x = -\frac{b}{2a}\), where \(a\) is the coefficient of \(x^2\) and \(b\) is the coefficient of \(x\).
In our function:
- \(a = -5\)
- \(b = 20\)
Now we can calculate the time:
\[ x = -\frac{20}{2 \cdot -5} = -\frac{20}{-10} = 2 \]
Therefore, the stone will reach its maximum height 2 seconds after being thrown.