SOLUTION: An NHL hockey arena holds 20�000 seats. 40% of the seats are in the upper bowl and 60% of the seats are in the lower bowl. At least 18�000 tickets are sold per game. A lower bowl t

Question 1104008: An NHL hockey arena holds 20�000 seats. 40% of the seats are in the upper bowl and 60% of the seats are in the lower bowl. At least 18�000 tickets are sold per game. A lower bowl ticket costs $85 and an upper bowl ticket costs $50. The management wants to maximize revenue.
a) define the variables
b) determine the restrictions on the variables
c) write a system of equations to represent each constraint:
I) number of seats in the upper bowl
II) number of seats in the lower bowl
III) number of seats sold per game
d) write the objective function that represents how revenue can be maximized for the variables.
Answer by greenestamps(13203) (Show Source): You can put this solution on YOUR website!

a) define the variables

Let x be the number of seats sold in the upper bowl;
let y be the number of seats sold in the lower bowl

b) determine the restrictions on the variables

Obviously both numbers of seats must be non-negative;
The maximum number of seats sold in the upper bowl can be no more than 40% of 20,000, or 8000;
The maximum number of seats sold in the lower bowl can be no more than 60% of 20,000, or 12000;
The total number of seats sold must be at least 18,000.

c) write a system of equations to represent each constraint:

Notes: (1) They are inequalities, not equations; and (2) you don't get a system of inequalities for each constraint; each constraint only produces one inequality. You can't "write a SYSTEM of inequalities" to represent EACH constraint.

I) number of seats in the upper bowl
There is no inequality for the number of seats in the upper bowl; the number of seats in the upper bowl is a constant, 8000.
The constraint for the number of seats in the upper bowl FOR WHICH TICKETS ARE SOLD is: 0 <= x <= 8000
II) number of seats in the lower bowl
There is no inequality for the number of seats in the lower bowl; the number of seats in the upper bowl is a constant, 12000.
The constraint for the number of seats in the lower bowl FOR WHICH TICKETS ARE SOLD is: 0 <= y <= 12000
III) number of seats sold per game: x+y >= 18000

d) write the objective function that represents how revenue can be maximized for the variables.

$50 per ticket sold in the upper bowl, plus $85 per ticket sold in the lower bowl: 50x+85y

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