calculate the equilibrium price of output in this industry

Firstly, you need to find out the market price (before the entering of the new firm) and quantity which is the intersection point between the demand curve nd supply curve:

24,000 - 600P = Qd = Qs = 4000 + 400P

That means 24,000 - 4,000 = 400P + 600P
20,000 = 1,000P
P = 20 (market price)
If P = 20, Qd =Qs = 4,000 + 400 X 20 = 4,800

It is a general principle of economics that a (rational) producer should always produce (and sell) the last unit if the marginal cost is less than the market price. As the market price will be dictated by supply and demand, it leads to the conclusion that marginal cost equals marginal revenue (= market price). These general principles are subject to a number of other factors and exceptions, but marginal cost and marginal cost pricing play a central role in economic definitions of efficiency. (Wikipedia)

When a new firm enter the market, we need to calculate how many output it will produce.

MC = P =10 + 0.5q
20 = 10 + 0.5q
q = (20 - 10) / 0.5 = 10 X 2 = 20

When 20 units of output from the new firm entering the market, there will be an increase in supply from 4,800 to 4,820. The market price tends to fall slightly as:

24,000 - 600P = Qd = Qs = (4000+20) + 400P
24,000 - 4,020 = 400P + 600P
19,980 = 1,000P
P = 19.98 (new market price)