The cyclist accelerates for 4 seconds.
Explanation:
The motion of the cyclist is a motion at constant acceleration, so we can use suvat equations:

where
s = 64 m is the distance covered
u = 15 m/s is the initial velocity
t is the time
is the acceleration
Substituting into the equation,

Solving the equation for t,

Which gives two solution: one is negative, so we discarde it since it has no physical meaning; the other one is
t = 4 s
which is the answer we were looking for.
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