Step 1 :The maximum height the diver reached can be found by finding the maximum point of the quadratic function \(h(t)=-4.9 t^{2}+3 t+10\). This can be done by finding the vertex of the parabola.
Step 2 :The vertex of the parabola is given by the formula \(t = -\frac{b}{2a}\) where \(a\) and \(b\) are the coefficients of \(t^2\) and \(t\) respectively.
Step 3 :Substituting the values of \(a\) and \(b\) into the formula, we get \(t = -\frac{3}{2*-4.9} = 0.3061224489795918\).
Step 4 :Once we have the value of \(t\) at the vertex, we can substitute it back into the function to find the maximum height.
Step 5 :Substituting \(t = 0.3061224489795918\) into the function \(h(t)=-4.9 t^{2}+3 t+10\), we get \(h = 10.4591836734694\).
Step 6 :Final Answer: The maximum height the diver reached is approximately \(\boxed{10.46}\) metres.