got my final for math tommorow and out of the review this is the only q i dont get:
A hockey arena sells premium tickets for $54. At this price, the arena will sell 150 premium tickets every game. The owners know from past years that they will sell 4 more premium tickets per game for each price decrease of $1. What price would let the owners own the same amount of money they earn now?
Can someone figure it out and explain please, thanks
A hockey arena sells premium tickets for $54. At this price, the arena will sell 150 premium tickets every game. The owners know from past years that they will sell 4 more premium tickets per game for each price decrease of $1. What price would let the owners own the same amount of money they earn now?
Can someone figure it out and explain please, thanks
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We first find out the amount of money the owners earn now.
Price per premium ticket = $54.
Total number of premium tickets that could be sold at this price = 150.
Hence, total earnings at this rate = 54*150 = 8100 ....................(1)
Now, it is given that every decrease of $1 implies an increase of 4 tickets.
So, if the owners decrease the price by x, then the increase in tickets would be 4x.
Applying these changes to the current price and current sale number, we have:
New ticket price = 54 - x
New (projected) number of tickets that could be sold = 150 + 4x
=> Total earnings with these changes = (54 - x)(150 + 4x) .................(2)
But the owners want to earn the same as they do currently, which is given by result (1).
=> (54 - x)(150 + 4x) = 8100
=> 8100 + 216x - 150x - 4x^2 = 8100
=> 66x - 4x^2 = 0
=> 2x(33 - 2x) = 0
=> x = 0 or 33 - 2x = 0
=> x = 0 or x = 33/2 = 16.5 ............... (3)
What result (3) means is that either a decrease of 0 dollars or a decrease of 16.5 dollars in the ticket price will result in the same earnings as result (1). We already know about the decrease of 0 dollars (i.e. no decreaes; no change in price). So we discard that solution.
Thus, the decrease in ticket price should be 16.5 i.e. the new ticket price = 54 - 16.5 = $37.5
This will result in an increase in the sale of 4*16.5 = 66 tickets, which takes the total of tickets sold to: 150 + 66 = 216.
Consequently, by a sale of 216 tickets at the rate of $37.5 per ticket, the owners will be able to earn:
216 * 37.5 = 8100 dollars, which is the same as they do now.
Solution: The ticket price that would let the owners earn the same amount of money they earn now is $37.5
Price per premium ticket = $54.
Total number of premium tickets that could be sold at this price = 150.
Hence, total earnings at this rate = 54*150 = 8100 ....................(1)
Now, it is given that every decrease of $1 implies an increase of 4 tickets.
So, if the owners decrease the price by x, then the increase in tickets would be 4x.
Applying these changes to the current price and current sale number, we have:
New ticket price = 54 - x
New (projected) number of tickets that could be sold = 150 + 4x
=> Total earnings with these changes = (54 - x)(150 + 4x) .................(2)
But the owners want to earn the same as they do currently, which is given by result (1).
=> (54 - x)(150 + 4x) = 8100
=> 8100 + 216x - 150x - 4x^2 = 8100
=> 66x - 4x^2 = 0
=> 2x(33 - 2x) = 0
=> x = 0 or 33 - 2x = 0
=> x = 0 or x = 33/2 = 16.5 ............... (3)
What result (3) means is that either a decrease of 0 dollars or a decrease of 16.5 dollars in the ticket price will result in the same earnings as result (1). We already know about the decrease of 0 dollars (i.e. no decreaes; no change in price). So we discard that solution.
Thus, the decrease in ticket price should be 16.5 i.e. the new ticket price = 54 - 16.5 = $37.5
This will result in an increase in the sale of 4*16.5 = 66 tickets, which takes the total of tickets sold to: 150 + 66 = 216.
Consequently, by a sale of 216 tickets at the rate of $37.5 per ticket, the owners will be able to earn:
216 * 37.5 = 8100 dollars, which is the same as they do now.
Solution: The ticket price that would let the owners earn the same amount of money they earn now is $37.5
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(150 + 4x)(54 - x) = 54(150)
-150x + 54(4)(x) - 4x^2 = 0
4x^2 - 66x = 0
x(4x - 66) = 0
x = 16.5
Price = 54 - 16.5 = 37.5
-150x + 54(4)(x) - 4x^2 = 0
4x^2 - 66x = 0
x(4x - 66) = 0
x = 16.5
Price = 54 - 16.5 = 37.5