# Maximize income and calculate maximum gross profit with service charge

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?

• math.stackexchange.com/questions/251768/… – lab bhattacharjee Jan 19 '17 at 6:33
• Your final answer is correct, but there is a constant missing in $-3x^2 + 120x = P$. – LinAlg Jan 19 '17 at 19:41
• @LinAlg No, my answer is wrong... – suomynonA Jan 20 '17 at 0:29
• I still disagree. The accepted answer assumes the question asks for the "gross profit if the price is not adjusted accordingly", while the current question clearly mentions the "maximum gross profit". – LinAlg Jan 20 '17 at 9:35