The relationship between the selling price of an item and the cost of an item has been determined to be q=-20s+400 has been determined by a marketing firm, where q is the quantity sold in a year and s is the selling price. If the cost to produce the item is $12, what is the selling price that optimizes the year yearly profit?

We are given with the variable cost which is:

q = - 20s + 400

The selling price is 's'. So, the profit can be represented by:

P = qs - q (12)

Subsituting:

P = (-20s + 400) s - 12 (-20s + 400)

P = - 20s^2 + 640s - 4800

To optimize this, we must differentiate the equation and equate it to zero, so:/

dP/ds = - 40s + 640 = 0

Solving for s,

s = 16

So, the selling price should be $16 to optimize the yearly profit.