A stadium can hold 25 000 people. People attending a regular event at the stadium must purchase a ticket in advance. When the ticket price is €20, the expected attendance at an event is 12 000 people. The results of a survey carried out by the owners suggest that for every €1 reduction, from €20, in the ticket price, the expected attendance would increase by 1000 people.
(a) If the ticket price was €18, how many people would be expected to attend?12000+(20−18)1000=14000
(b) Let x be the ticket price, where x ≤ 20. Write down, in terms of x, the expected attendance at such an event. \[12000+(20− x)1000= 32000−1000x\]
(c) Write down a function f that gives the expected income from the sale of tickets for such an event. \[f(x) = (32000−1000x)x\]
(d) Find the price at which tickets should be sold to give the maximum expected income. \[f(x) = (32000−1000x)x\] \[f(x) = 32000−2000x = 0\]... \[x = €16\]
(e) Find this maximum expected income. \[f(x) = (32000−1000x)x\]\[f(16) = (32000−16000)16 = €256000\]
(f) Suppose that tickets are instead priced at a value that is expected to give a full attendance at the stadium. Find the difference between the income from the sale of tickets at this price and the maximum income calculated at (e) above. \[32000 −1000x = 25000\]...\[1000x = 7000\]...\[x = 7\] \[f(x) = (32000−1000x)x\]... \[f(7) =(32000−7000)7 =175000\] Difference: €256 000 − €175 000 = €81 000
(g) The stadium was full for a recent special event. Two types of tickets were sold, a single ticket for €16 and a family ticket (2 adults and 2 children) for a certain amount. The income from this event was €365000. If 1000 more family tickets had been sold, the income from the event would have been reduced by €14000. How many family tickets were sold? x = number of single tickets f = number of family tickets y = cost of family ticket x+4f = 25000\[16x+fy= 365000\]\[16(x-4000)+(f+1000)y = 351000\]\[16x - 64000+fy+ 1000y = 351000\]\[16x + fy = 365000\]__________________\[1000y=50000\]\[y=50\]\[x+4f = 25000\]\[16x+ 50f = 365000 16x+ 64f = 400000\]________________\[14f = 35000\]\[f=2500\]