The same expression $800 - 10p$ appears in both the definition of $n$ and the revenue function $R(p)$. It should represent the same thing in both places.

Revenue from ticket sales is calculated using two quantities multiplied together. One is the ticket price. What is the other?

In $R(p) = p(800 - 10p)$, you know $p$ is the ticket price. So consider what $800 - 10p$ must be so that $p \times (800 - 10p)$ gives a revenue.

In Desmos, type n = 800 - 10p on one line and R(p) = p(800 - 10p) on another line. This lets you see that the same expression, 800 - 10p, appears in both equations.

Note that in $R(p) = p(800 - 10p)$, the factor p is already defined in the problem as the ticket price. In Desmos, you can create a slider for p to see how changing the price changes n and R(p), and think about what 800 - 10p must represent so that p * (800 - 10p) gives a revenue amount.

The problem tells you that the number of tickets that can be sold at price $p$ is modeled by

So, in words, the expression $800 - 10p$ is exactly the same thing as $n$ in this context.

Revenue is defined as

Revenue from ticket sales is always

• (ticket price) $\times$ (number of tickets sold).

Here, $p$ is already defined as the ticket price, so you know one factor represents the price.

Since $R(p)$ is price times quantity, and $p$ is the price per ticket, the other factor, $800 - 10p$, must represent the quantity part of the product, which is the number of tickets sold.