a sudbrook school play has a maximum of 700 tickets to sell. Students sellling the tickets inn adavance. at least 300 tickets must be sold in advance. the rest of the tickets are avaliable for sale at the play. Advance tickets will be sold for $20.oo while the tickets bought at the door will cost $30.00.
Write a system of equalities
Graph the feasible region
Determine the vertices
How many of each ticket should be sold to maximize the profit?
Number of advance tickets =x
Number of tickets sold at the play=y
Total number of tickets:
At least 300 tickets must be sold in advance:
x`>=` 300 then y`<=` 400
The total proceeds are equal to 20x+30y. Total proceeds need to be maximized. The maximum profit would be when x is the smallest, x=300, so y=700-300=400,
The maximum profit is (20)(300)+(30)(400)=6000+12000=18000
Graph the three lines:
The feasible region is the triangle formed by x=300, the x axis and the y=700-x line.
The vertices are (300,0), (300,400) and (700,0). These poins represent
(300,0): Only 300 tickets are sold in advance, no tickets are sold at the play, then the profit is = 300x20=6000
(700,0): 700 tickets are sold in advance, thus the profit is = 700x20=14000
(300,400): 300 tickets are sold in advance, 400 at the play, then the profit is the highest = 18000
Thus 300 tickets must be sold in advance and 400 at the play in order to maximize the profit.