1. **Problem:** Find the maximum height of the cannonball given by the equation $$h = -16t^2 + 128t$$. 2. **Formula:** The height function is a quadratic in the form $$h = at^2 + bt + c$$ where $$a = -16$$, $$b = 128$$, and $$c = 0$$. 3. **Maximum height:** Since $$a < 0$$, the parabola opens downward and the vertex represents the maximum height. 4. The time at which the maximum height occurs is given by $$t = -\frac{b}{2a}$$. 5. Substitute values: $$t = -\frac{128}{2 \times -16} = -\frac{128}{-32} = 4$$ seconds. 6. Find the maximum height by substituting $$t=4$$ into the height equation: $$h = -16(4)^2 + 128(4) = -16(16) + 512 = -256 + 512 = 256$$ feet. **Final answer:** The maximum height of the cannonball is **256 feet**.