To find the equilibrium output and profit for a monopolist, we need to set the marginal cost equal to the marginal revenue.

The marginal cost is the additional cost of producing one more unit of output, and the marginal revenue is the additional revenue generated by selling one more unit.

Given that the production function is ( q = 2\sqrt{X} ), we can find the marginal cost (( MC )) by taking the derivative of the production function with respect to ( X ):

[ MC = \frac{dq}{dX} ]

[ MC = \frac{d}{dX} (2\sqrt{X}) ]

[ MC = \sqrt{X} ]

Now, we can find the inverse demand function by solving for ( q ) in terms of ( P ) from the demand function ( P = 85 – 3q ):

[ q = \frac{85 – P}{3} ]

The total revenue (( TR )) is the product of the price (( P )) and the quantity demanded (( q )):

[ TR = P \cdot q ]

[ TR = P \cdot \left(\frac{85 – P}{3}\right) ]

Now, find the marginal revenue (( MR )) by taking the derivative of the total revenue with respect to ( q ):

[ MR = \frac{dTR}{dq} ]

[ MR = \frac{d}{dq} \left(P \cdot \left(\frac{85 – P}{3}\right)\right) ]

[ MR = \frac{85 – 2P}{3} ]

Now, set the marginal cost equal to the marginal revenue to find the equilibrium output:

[ \sqrt{X} = \frac{85 – 2P}{3} ]

Next, we substitute the demand function ( q = \frac{85 – P}{3} ) for ( X ):

[ \sqrt{\frac{85 – P}{3}} = \frac{85 – 2P}{3} ]

Square both sides to solve for ( P ):

[ \frac{85 – P}{3} = \left(\frac{85 – 2P}{3}\right)^2 ]

Solving this equation will give us the equilibrium price (( P )). Once we have ( P ), we can substitute it back into the demand function ( q = \frac{85 – P}{3} ) to find the equilibrium quantity.

Finally, we can calculate the profit by subtracting the total cost from the total revenue:

[ Profit = TR – TC ]

The total cost (( TC )) is the product of the quantity produced and the fixed cost per unit (( X \cdot 5 )).

Please note that the specific numerical solution will depend on the values obtained for ( P ) and ( q ) from solving the equations.