at time t, and we are discussing 1 dimensional motion.
When the rocket is at its maximum height its velocity is 0.
The velocity is obtained by differentiating the equation for the height, so
the velocity is:
df/dt = 24.5 - 9.8t.
At the maximum height df/dt=0, so this occurs when:
24.5 - 9.8t = 0,
t = 24.5/9.8 ~= 2.5 seconds
So the fuse would have to be set to 2.5 seconds to have the rocket explode
at its maximum height.
The height at which the rocket explodes is:
f(2.5) = 24.5*2.5 - 4.9*2.5^2 = 30.625 metres.