h(t) = -4.9t^2 + 30t + 53

a) ball goes up, comes back down to the top of the tower. So, we want

53 = -4.9t^2 + 30t + 53

0 = -4.9t^2 + 30t

0 = t(-4.9t + 30)

so, t=0 (at the start) or t = 6.12 (as it comes back down)

If t(-4.9t+30)=0, either

t=0

or

-4.9t+30 = 0

That is, t = 6.12

If you can't solve a factored expression, you have some review to do.

b) same calculation is between 0 and 6.12. That is 0 < t < 6.12

c) same calculation, but restricting t to positive values, t>6.12

Naturally, we could also restrict t to the point where height >= 0.

d) vertex of any parabola is where x = -b/2a = -30/-9.8 = 3.06

you know from the quadratic formula that x = -b/2a ± sqrt(blah blah)

Parabolas are symmetric, so the vertex is midway between the roots, which are equally spaced around x = -b/2a

h(3.06) = 98.9