SOLUTION: A fruit juice company varies the price of its litre cartons to maximise profit. The equation that models the company's profit is given below: P=-80d^2+960d-2280 Where P is the pr

Question 1144637: A fruit juice company varies the price of its litre cartons to maximise profit. The equation that models the company's profit is given below:
P=-80d^2+960d-2280
Where P is the profit in thousands of dollars and d is the price of a litre carton in dollars.
a) What is the maximum profit the company can make selling litre containers?
b) Sketch the general shape of the graph showing how profit varies with the price of a litre container.
c) What's the lowest price the company can sell litre cartons for and still break even? (Answer to the nearest 10 cents)
Answer by ikleyn(52780) (Show Source): You can put this solution on YOUR website!
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(a)  Function  P(d) = - 80d^2 + 960d - 2280  is a quadratic function of the argument (variable) "d".


     For any quadratic function  f(x) = ax^2 + bx + c  with negative leading coefficient "a" at  x^2, 

     its maximum is achieved at the value of   = .


     In your case,  a= -80,  b= 960.  Therefore

          =  =  = 6.


     
      Thus the optimum price is  6 dollars (or whatever monetary units) per cartoon.

      It provides the maximum profit of  P(6) = -80*6^2 + 960*6 -2280 = 600 thousands of your monetary units.



(b)  Make a plot on your own.  //  When a "student" asks me to make a plot, I always think that it is his (or her) job - not my. 



(c)  To answer this question, solve this quadratic equation


         P(d) = 0,  or  -80*d^2 + 960*d -2280 = 0.

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On finding the maximum/minimum of a quadratic function see the lessons in this site
    - HOW TO complete the square to find the minimum/maximum of a quadratic function
    - Briefly on finding the minimum/maximum of a quadratic function
    - HOW TO complete the square to find the vertex of a parabola
    - Briefly on finding the vertex of a parabola

    - A rectangle with a given perimeter which has the maximal area is a square

    - OVERVIEW of lessons on finding the maximum/minimum of a quadratic function

Also, you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this textbook under the topic "Finding minimum/maximum of quadratic functions".

Save the link to this online textbook together with its description

Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson

to your archive and use it when it is needed.

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