>>> I don't like VERY MUCH when somebody comes for help and starts teach me how to solve the problem.

         >>> Especially when your advice makes the solution harder . . . 

         >>> Let me decide how to do it better - or solve the problem on your own . . . 

     The phrase " . . . for every dollar increase, it loses 3 sales per week." means that at the price (9+x) dollars the number of sales is (30-3x) feeders per week.


     In  turn, it means that the profit function per week as the function of x  is 


        P(x) = (9+x)*(30-3x) - 5*(30-3x)   (1)   (profit = revenue minus cost)


     It is the answer to question (a).

     

(b)  What price should the society charge for each feeder to maximize profits?


      All you need is to simplify (1) and find the maximum of this quadratic function:


      P(x) = 270 + 30x - 27x - 3x^2 - 150 + 15x = -3x^2 + 18x + 120.


      For a quadratic function q(x) = ax^2 + bx + c   with negative leading coefficient "a" the maximum is achieved at x = .  

      In your case  a = -3,  b = 18,  so  x =  = 3.


      So, the optimal price is  9+3 = 12 dollars  per feeder.



(c)  What is the maximum weekly profit?


     The current weekly profit is  30*$9 - 30*5 = 120 dollars.


     The projected (predicted) weekly profit is  (see formula (1) )


        (9+3)*(30-3*3) - 5(30-3*3) = 147 dollars.