A baseball team plays in a stadium that holds $\displaystyle 55,000$ spectators. With the ticket price at $\displaystyle $10$, the average attendance at recent games has been $\displaystyle 27,000$. A market survey indicates that for every dollar the ticket price is lowered, attendance increases by $\displaystyle 3000$.

(a)Find a function that models the revenue in terms of ticket price. (Let $\displaystyle x$ be the ticket price and $\displaystyle R(x)$ be the revenue.)

So far, I have the following (but can't put it all together):

$\displaystyle Revenue=attendance * price$

$\displaystyle attendance = avg. attendance + 3000*(price reduction)$

$\displaystyle t=10$: tick price

$\displaystyle p$: price reduction

$\displaystyle x$: attendance

$\displaystyle R(x)$: Revenue in terms of attendance

$\displaystyle R(x)= xt=10x$

$\displaystyle x=27000+3000p$

$\displaystyle R(x)=270000+30000p$

Wrong letter ^ ^

How do I relate $\displaystyle t$ and $\displaystyle p?$

$\displaystyle 3000*(t-p)$ gives a smaller number for a bigger $\displaystyle p$ (wrong).

$\displaystyle 3000*(p-t)$ gives negative numbers (for example $\displaystyle p=1$ is a $\displaystyle $1$ reduction means $\displaystyle p-t=-9$), but at least it gets bigger as $\displaystyle p$ gets bigger.

I could do these two myself if I could do (a).

(b) Find the price that maximizes revenue from ticket sales.

(c) What ticket price is so high that no revenue is generated?