"I thought that I read something somewhere that said that marginal cost is equal to price."

I think the book says that Marginal Cost = supply price in a short-run competitive market.  Thus in order to maximize profit the price must equal marginal cost.  In parts B and C I got a shutdown price of $5 occuring at a quantity of 5.  The table says that at a quantity of 5, var_cost=25 and fix_cost=10 for a total cost of 35.  Part C says that at the shutdown price, the total revenue = total variable cost (which is 25 on the according to the table).

"but I think the shutdown price would be where the quantity is equal to 5.  That would make the price 4 and TR = 4*5 = 20"

I agree that the shutdown price occurs at a quantity of 5, but why is the price = $4?  I get a shutdown price also of 5.  So I get a total revenue of $25 and a profit of -10.  Di I miss something?

I am not certain how you got -80, and I could be very wrong, but I think the shutdown price would be where the quantity is equal to 5.  That would make the price 4 and TR = 4*5 = 20  subtract from that a total cost of $35 and you get -15.