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A concert is soon holding and the venue can hold 12000 people. The minimum price is 100 and all tickets can be sold out if the price of each ticket is set to this minimum. For every increment of 1 in the price, the number of tickets sold decreases by 20. It is given that the total expenditure of the performance is 1000000. Let x be the price of each ticket and $y be the overall profit of the concert.

 

(a) Show that the number of tickets sold at $x is 14000-20x.

(b) Show that y=-20x^2 +14000x -1000000.

(c) Write down the domains of the function in (a).

(d) Find the range of prices of each ticket such that there is an overall profit.

(e) Find the price of each ticket such that the profit is at maximum and the number of ticket sold at this price.

Thanks

 Dec 20, 2019
 #1
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Assuming that the  venue can hold  14,000 people  ( not,  12,000 people)

 

(a)  Let x  be the number of incremental increases

 

So  the number of tickets sold  = (14000 - 20x)

 

(b)

 

We have that

 

Profit  =  Revenue  -    Cost of staging concert

 

Revenue   =  Number of people  *  Price 

 

Let x be the number of   incremental price increases......so....

 

Revenue  =  (14000 - 20x) ( 100 + x)      simplify

 

Revenue =  1400000 - 2000x + 14000x - 20x^2  =  -20x^2 + 12000x  + 1400000 

 

So.....the total profit is given by

 

Revenue - Expenses...so.....

 

y = -20x^2 + 12000x + 1400000 - 1000000

 

y= -20x^2 + 12000x  + 400000

 

(c)   See the graph, here : https://www.desmos.com/calculator/kqdbrpbuuz

The domain is  [0, ≈ 632 ]   

 

(d) The range of prices that  will guarantee a profit =

 

(100 + 0 , ≈ 100 + 632)  =    [100, ≈732 ]    (in dollars)

 

(e)  Looking at the graph again.......the max occurs at  (300, 2,200,000)

 

So  the price that will guarantee a max =  (100 + 300)  = $400

 

And the  number of tickets sold at this price =  14000 - 20(300)  = 8000 tickets

 

 

cool cool cool

 Dec 20, 2019

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