A $300$-room hotel is filled to capacity every night at $\$80$ a room. For each $\$1$ in rent, $3$ fewer rooms are rented. How much should the management charge for each room to maximize income? If each room rented costs management $\$20$ service per day, what is the maximum gross profit?
For the question, "How much should the management charge for each room to maximize income," I obtained the correct answer:
$$(80+x)(300-3x)=P$$ $$-3x^2+60x+24000=P$$ $$x=\dfrac{-60}{-6}=10$$ $$\text{Management should charge}\; \$90$$.
For the question, "If each room rented costs management $\$20$ service per day, what is the maximum gross profit," my answer of $\$19200$ was incorrect. The correct answer was $\$18900$. My work:
$$(60+x)(300-3x)=P$$ $$-3x^2+120x+18000=P$$ $$x=\dfrac{-120}{-6}=20$$ $$80\cdot240=\$19200\;\text{as the maximum gross profit}$$
What did I do wrong?
I think you only need to calculate how many rooms will be rented at $90 per night.
(rate - service charge) × number of rooms