Based on past events, a charity fundraising carnival predicts that it will sell 2400 tickets if the ticket price is \$8. The break-even ticket price is \$8, so tickets must not be priced below this. Ticket prices are set in whole dollars.
The organising committee estimates that for every \$1 increase in the ticket price, 100 fewer tickets will be sold.
Complete the table below.[1]
| Increase in ticket price, x (\$) | New ticket price (\$) | Expected number of tickets sold | Revenue, R (\$) |
|---|
| 0 | 8 | 2400 | 19 200 |
| 1 | 8+1=9 | 2400−100(1)=2300 | 20 700 |
| 2 | 8+2=10 | 2400−100(2)=2200 | 22 000 |
| \(x\) | \(8+x\) | \(2400-100x\) | |
If the committee wishes to collect exactly \$23 100 from ticket sales, while maintaining an attendance of at least 2000 people, show that the ticket should be priced at \$11.[4]
If however, the organisers wish to maximise revenue, find the ticket price they should set, and state this maximum revenue.[2]
Two corporate sponsors have each offered to support the charity, but the committee may accept only one of them:
Sponsor A will donate \$15 000 if at least 1900 tickets are sold.
Sponsor B will donate fifty cents for every dollar collected from ticket sales.
Determine which sponsor the committee should accept, and the ticket price it should set, in order to raise the greatest total amount for the charity. Justify your recommendation with calculations.[4]