Question

A charity organization had a fundraiser where they sold each ticket for a fixed price. After selling
\[200\] tickets, they had a net profit of
\[\$12{,}000\]. They had to sell a few tickets just to cover necessary production costs of
\[\$1{,}200\].
Let
\[y\] represent the net profit (in dollars) when they have sold
\[x\] tickets.
Complete the equation for the relationship between the net profit and number of tickets sold.
\[y=\]

Answer 1

To determine the relationship between the net profit \(y\) and the number of tickets sold \(x\), we start by noting two key pieces of information:

1. The charity organization had a net profit of \(\$12,000\) after selling \(200\) tickets.
2. They have production costs of \(\$1,200\) which must be covered before any profit can be made.

Let's denote:
- \(p\) as the price of each ticket.
- \(C\) as the total production costs, which is \(\$1,200\).

The total revenue from selling \(x\) tickets is given by:
\[
\text{Revenue} = p \cdot x
\]

The net profit \(y\) can be expressed as the total revenue minus the production costs:
\[
y = \text{Revenue} - C
\]
Substituting for Revenue, we have:
\[
y = p \cdot x - 1200
\]

Now, we also know that when \(x = 200\), \(y = 12000\):
\[
12000 = p \cdot 200 - 1200
\]

Now we can solve for \(p\):
\[
12000 + 1200 = p \cdot 200
\]
\[
13200 = p \cdot 200
\]
\[
p = \frac{13200}{200} = 66
\]

Now we can substitute \(p\) back into the equation for \(y\):
\[
y = 66x - 1200
\]

Thus, the complete equation for the relationship between the net profit \(y\) and the number of tickets sold \(x\) is:
\[
y = 66x - 1200
\]