Maximizing Profit and Optimization

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? I understand that the profit function can be found through taking the revenue function minutes then cost functions and I get the revenue function by multiplying the demand function by the unit but what do I do then? And how do I know how many items are sold at the optimizing price?

Answer 1

Let us start with some assumptions.

If x is the quantity of goods produced and p is the price per each good we have the following:

C(x) = 2x + 45

Q(p) = 10- p^(1/2)

The cost function is unbounded, given it is a linear function, ie the more goods you make the more it will cost you. The quantity however, as the price will be between 0 and 100.

Now with that said, we conceptually must remember that revenue is quantity demanded multiplied by the price.

R = Q(p)p = 10p - p^(3/2)

Profit is maximized at the point where the first derivative is equal to 0.

R' = 0 = 10 - p^(1/2)*3/2

Using algebra to solve for p, one would get the maximum price as 400/9, which is slightly more than 40 dollars (I am assuming this is what currency you are using). Going back, one could solve for the quantity and the cost at said quantity to verify this is a maximum profit point. This should answer your question. I hope a final exam did not depend on it. Please message me or book an appointment if you have further questions.