Maximum profit.

Question

 Maximum profit. A chain store manager has been told by the main office that daily profit, P, is related to the number of clerks working that day, x, according to the function P = −25x2 + 300x. What number of clerks will maximize the profit, and what is the maximum possible profit? Step by step answers please.

Answer 1

This is the equation of a parabola that opens downwards (because the coefficient of x² is negative) so the maximum value of P will be found at the parabola's vertex.

 

The x-coordinate of the vertex is given by:   -b

                                                                 2a

 

Your equation is already in the standard form: (a = -25, b = 300, c = 0) so we can find the x-coordinate at which P will be the maximum.

 

- 300    =   -300     =   6  So six clerks will maximize the profit.

2(-25)        -50

 

To find the maximum profit, plug 6 into the equation for X

 

P = -25 (6)² + 300(6)

    

   = -25 (36) + 1800

 

   =  -900 + 1800

 

  =  900  this is the maximum profit