Question 12 (2 points) A football is punted into the air. Its height is given by $h(t)=-4.9 t^{2}+24.5 t+1$ where $h$ is in meters and $t$ is in seconds Find the maximum height of the ball and the time the ball reaches its maximum height

Step 1 ：The height of the ball is given by a quadratic function. The maximum height of the ball is the vertex of the parabola represented by the quadratic function. The vertex of a parabola given by \(f(x) = ax^2 + bx + c\) is at \(x = -\frac{b}{2a}\). In this case, \(a = -4.9\) and \(b = 24.5\). So, the time at which the ball reaches its maximum height is \(t = -\frac{b}{2a}\).

Step 2 ：Substituting this value of \(t\) into the height function will give the maximum height.

Step 3 ：Given \(a = -4.9\), \(b = 24.5\), and \(c = 1\), we find that \(t = 2.5\) and \(h = 31.624999999999996\).

Step 4 ：Final Answer: The ball reaches its maximum height of approximately \(\boxed{31.625}\) meters at \(\boxed{2.5}\) seconds.