(See attached file for full problem description)
Part A: I attempted to find the profit-maximizing output and I got 42.857. From there I came up with a price of -25.71 (I don't feel that this is correct). After that I tried to find the profit, but when I went back to substitute my quantity (Q1) back into the profit equation I had a Q and I don't have a value for Q.
Here are your answers.
Here, we must first write the profit function. Revenue is P*Q (where P is given by the demand function) and cost is given by the cost function. Therefore, the profit function is:
Profit = P*Q - C = (660 - 16Q)Q - 900 - 60Q - 9Q^2 = 660Q - 16Q^2 - 900 - 60Q - 9Q^2
[the ^ symbol means "to the power of"]
In order to find the profit-maximizing quantity, we find the derivative of this function and equate it to zero. The derivative with respect to Q is:
660 - 32Q -60 - 18Q = 600 - 50Q
When equating it to zero, we get:
600 - 50Q = 0
50Q = 600
Q = 12
So the quantity is 12. Plugging this value into the demand function, we get that the price will be P = 660 - 16*12 = $468. Finally, the profit can be found by plugging Q=12 into the profit function we found above.
Profit = 660*12 - 16(12)^2 - 900 - 60*12 - 9(12)^2 = $2,700
So profit will be ...