If I know the demand function of a new product is q(p)=10 - the square root of p, and I know that the cost function is C(x)=2x+45, how do I find the unit price that maximizes profit? What is the maximum profit? And how many items are sold at the optimizing price?

Question

If I know the demand function of a new product is q(p)=10 - the square root of p, and I know that the cost function is C(x)=2x+45, how do I find the unit price that maximizes profit?  What is the maximum profit?  And how many items are sold at the optimizing price?

oobleck · Answer

Not quite sure how p,q,x are related, but profit = revenue - cost and revenue r = price * quantity So, if you can sort those out, find where dr/dp = 0 and find q at that price p

Blare · Answer

So from what I have been told, q is the demand function and that can be used to find the revenue function by multiplying it by p. I’m not sure how to plug in but I plug in to the quadratic somehow. I know that the unit price that maximizes profit is at 45.83. And items sold at the optimizing profits is 3.23. The only thing I don’t understand how to find now is the maximum profit.