If p is the angle of projection then the vertical component of the velocity is u=50sin(p).
The equation of vertical motion is height y=ut-½gt² where g is the acceleration of gravity=9.8m/s² approx. The equation can be rewritten:
y=ut-4.9t²=-4.9(t²-ut/4.9).
We can complete the square:
y=-4.9(t²-ut/4.9+u²/9.8²-u²/9.8²),
y=-4.9[(t-u/9.8)²-u²/9.8²],
y=u²/19.6-4.9(t-u/9.8)².
The maximum value of the quadratic expression is u²/19.6=100 because 100m is the maximum height. Substituting for u: 2500sin²(p)/19.6=100, sin(p)=√(1960/2500)=√0.784=0.8854 approx, making p=62.31° approx.