This is a very good question, as it highlights the dynamic nature of friction in physics.
Static friction, fundamentally, is a force that acts on two objects that are right up against each other, but whose surfaces aren't sliding against each other (that is, their surfaces have a velocity of 0 relative to each other). It applies just enough force to ensure they continue having a relative surface-to-surface velocity of 0, no more, no less. If that force exceeds a certain amount, μs*Normal force, then it stops applying and lets them start sliding against each other.
In this case, static friction needs to apply a force towards the center of the rotating disk (it acts as centripetal force) to prevent the person from sliding in a straight line off the disk. The formula for that is F=mv²/r. In this case, that would be 80*v²/5.
Meanwhile, the maximum force of static friction would be the normal force times the static friction. I'm going to assume the rotating platform is completely horizontal. Since the only vertical forces are gravity and the normal force, the normal force must be equal and opposite to gravity to prevent the person from falling through the floor or floating off into space. So N = mg = 80*g. And maximum static friction = 80*g*0.45.
So, the maximum tangential speed would be the one such that the maximum static friction is also the required static friction to prevent slipping.
80*g*0.45 = 80*v²/5
0.45g = v²/5
I'm going to assume the gravitational constant you're using in this class is 9.8.
9.8*0.45 = v²/5
5*9.8*0.45 = v²
v = √(5*9.8*0.45)
v = 4.70 m/s
The maximum tangential velocity is 4.70 m/s
I can't draw a free body diagram because that's not how this platform works.
Also, as a bonus, the rotational velocity would be 4.70m/s / 5m = 0.94 radians per second, but that wasn't part of the question.
