Answer:
The angular velocity is
Explanation:
From the question we are told that
The height of Luc is 
The weight of Luc is 
The mass of the dumbbell is 
The diameter of the merry-go-round is 
The diameter of the merry-go-round is 
The distance of the hand from the center is 
The diameter of Luc is 
The diameter of the merry-go-round is 
The weight of the merry-go round is 
The steady speed of the merry go round is 
Generally the moment of inertia before Luc brings his hand to his chest is mathematically represented as
Where
is the mass of the merry-go round which is

is the mass of Luc which is

is the mass of dumbell which is

So


The moment of inertia after Luc brings his hand to his chest is mathematically represented as

Substituting value


According to the law of conservation of angular momentum


Substituting values

Converting back to rpm