To determine how much money the charity raises (or loses) by selling 12 tickets, given the profit function:
[tex]\[ P = 70n - 1500 \][/tex]
where [tex]\( P \)[/tex] is the profit and [tex]\( n \)[/tex] is the number of tickets sold.
We know that:
[tex]\[ n = 12 \][/tex]
We substitute [tex]\( n = 12 \)[/tex] into the profit equation:
[tex]\[ P = 70 \times 12 - 1500 \][/tex]
Given the result from the solution already provided, we know that:
[tex]\[ P = -660 \][/tex]
This means the charity has a negative profit (loss) of [tex]$660 when 12 tickets are sold.
To find out how much money the charity raises by selling a single ticket, we need to understand the given model's relationship:
Each ticket sold contributes $[/tex]70 to the income, while the fixed costs (expenses) for the fundraiser are [tex]$1500. Hence, the profit function is:
\[ P = 70n - 1500 \]
From the calculation above, we see that selling 12 tickets results in a loss of $[/tex]660. Therefore, it is clear that the fundraising expenses exceed the income generated from ticket sales, specifically:
For [tex]\( n = 12 \)[/tex]:
[tex]\[ P = 70 \times 12 - 1500 \][/tex]
[tex]\[ = 840 - 1500 \][/tex]
[tex]\[ = -660 \][/tex]
So the income from selling each ticket ([tex]\( n \)[/tex]) does indeed bring in $70 per ticket, but due to high expenses, there's still a loss with the sale of 12 tickets.
