SOLUTION: Gemma's mother owns a manufacturing company that produces key rings. Last year, she collected data about the number of key rings produced per day and the corresponding profit. The

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Question 1183271: Gemma's mother owns a manufacturing company that produces key rings. Last year, she collected data about the
number of key rings produced per day and the corresponding profit. The data can be modeled by the relation
𝑷 = βˆ’πŸπ’Œ^𝟐 + πŸπŸπ’Œ βˆ’ 𝟏𝟎, where 𝑷 is the profit in thousands of dollars and π’Œ is the number of key rings in
thousands. You must solve the following questions by factoring.
a) How many key rings must be produced so that there is no profit and no loss [𝑷 = 𝟎]?
b) How many key rings must be produced for the maximum profit?
c) What is the maximum profit?
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.
Gemma's mother owns a manufacturing company that produces key rings. Last year, she collected data about the
number of key rings produced per day and the corresponding profit. The data can be modeled by the relation
𝑷 = βˆ’πŸπ’Œ^𝟐 + πŸπŸπ’Œ βˆ’ 𝟏𝟎, where 𝑷 is the profit in thousands of dollars and π’Œ is the number of key rings in
thousands. You must solve the following questions by factoring.
a) How many key rings must be produced so that there is no profit and no loss [𝑷 = 𝟎]?
b) How many key rings must be produced for the maximum profit?
c) What is the maximum profit?
~~~~~~~~~~~~~~ 

(a)  To answer, you should solve the equation  P = 0, which is

         -2k^2 + 12k - 10 = 0,

     or, EQUIVALENTLY,  

         k^2 - 6k + 5 = 0.


     Factor

         (k-5)*(k-1) = 0

     and find two roots,  5 and 1.


     This question has two answers:  1 thousand keys and 5 thousand keys.



(b)  and  (c)   The maximum profit corresponds to the vertex of the quadratic function.

         -2k^2 + 12k - 10 = -2*(k^2 - 6k + 5) = -2*((k-3)^2 - 4).

      The vertex is at  k= 3,  with the value of the quadratic function equal to  (-2)*(-4) = 8.


      THEREFORE, the answer to question (b) is 3000 keys (the optimal production) and  8000 dollars (the maximum profit).

Solved.     //     All questions are answered.



Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


a) Solve P=0:




or

ANSWERS: The profit is 0 when k is 1 or 5 -- that is, when either 1000 or 5000 key rings are produced per day.

b) Using the symmetry of the parabolic graph, the maximum profit is when k is halfway between the two zeros of 1 and 5 -- i.e., when k=3.

ANSWER: The maximum profit is when 3000 key rings are produced per day.

c) The maximum profit is P(3):



ANSWER: The maximum profit is 8000 dollars
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