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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?

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  • $\begingroup$ math.stackexchange.com/questions/251768/… $\endgroup$ – lab bhattacharjee Jan 19 '17 at 6:33
  • $\begingroup$ Your final answer is correct, but there is a constant missing in $-3x^2 + 120x = P$. $\endgroup$ – LinAlg Jan 19 '17 at 19:41
  • $\begingroup$ @LinAlg No, my answer is wrong... $\endgroup$ – suomynonA Jan 20 '17 at 0:29
  • $\begingroup$ 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". $\endgroup$ – LinAlg Jan 20 '17 at 9:35
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I think you only need to calculate how many rooms will be rented at $90 per night.

(rate - service charge) × number of rooms

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