Calculus question

Question

If the price of each ticket to a concert is $16, a total of 4000 tickets will be sold. For each $1 increase in price, ticket sales will decrease by 100. What should the ticket price be set at in order to maximize revenue?

Answer 1

 

 Demand function maps the price to the quantity demanded.

 

  At $16 the demand is 4000.

  For each $1 increase in price the demand decreases by 100

 

  So the ordered pairs are (16, 4000)  

                                      (17, 3900)

                                      (18, 3800)

                                      (19, 3700)

                                      (20, 3600)

   etc.

 

 The slope is -100.

                               at $56, the demand will be zero.

                                free tickets, the demand will be 5600

                              

     The equation of the demand line is y = -100x + 5600 = P(x)

 

The revenue function is R(x) = x*P(x) = x( -100x + 5600)

                                                       

                                                        = -100x^2 + 5600x

 

  R'(x) = dR/dx = -200X + 5600

 

           Maximizing:  0 = dR/dx = -200X + 5600

 

                            -5600 = -200X

 

                                X = 28