1. ## Calculus Word Problems

A hotel has 72 rooms.
Rooms are usually $90 per night. But if the owner raises the price to$95 per room per night, he finds that there will be 2 rooms fewer than usual($90 a room per night) hired. I am not sure whether for EVERY$95 dollar room, 2 rooms are left vacant, or are 2 rooms of the 72 rooms in total left vacant.

I think our teacher said that the Income formula that we have to calculate is:
Income = 108x - [or maybe +] (2x^2)/5
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1. Find the formula to calculate the income

2. Calculate the maximum Income

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Could someone please explain calculus word problems like this to me in detail?

Thank You!

2. Originally Posted by janvdl
A hotel has 72 rooms.
Rooms are usually $90 per night. But if the owner raises the price to$95 per room per night, he finds that there will be 2 rooms fewer than usual($90 a room per night) hired. ... 1. Find the formula to calculate the income 2. Calculate the maximum Income Hello, I assume: a) that each room is rented for$90
b) that the price could be increased by multiple of $5: let x be the times that the price is increased: let i be the income: i(x) = (72 - 2x)(90 - 5x) = -10x² + 180x + 6480 The original situation is i(0) =$6480

The graph of this function is a parabola opening downward so the maximum value is at it's vertex. Use derivation:
i'(x) = -20x + 180. You'll get the maximum if i'(x) = 0 ==> x = 9

i(9) = $7290 when only 72 - 18 = 54 rooms are rented for a price of$(90-5*9) = $45 3. Sorry Earboth, my mistake. You are correct in what you did, but it still doesnt give me the income formula that I should have got... 4. Hello, janvdl! The problem is badly worded, but I'm familiar with this type. A hotel has 72 rooms. .Rooms are usually$90 per night.
But if the owner raises the price to $95 per room per night, he finds that there will be 2 fewer rooms rented than usual. (1) Find the formula to calculate the income (2) Calculate the maximum Income The problem is usually stated like this . . . When the owner charges$90 per room, all 72 rooms are rented.
Then he finds that for every $5 increase in the rate, he loses 2 guests. Let x = number of five-dollar increases in the rate. The rate will be: .90 + 5x dollars per night. He will lose 2 guests for every increase. . . Hence, he will have only: .72 - 2x guests. His income will be: .(90 + 5x)(72 - 2x) dollars. Therefore, the income function is: .I .= .6480 + 180x - 10x² .(1) Maximize the income function: .I' = 0 We have: .180 - 20x .= .0 . . . . x = 9 .** The maximum income is: .I .= .6480 + 180·9 - 10·9² .= .$7290 .(2)

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To maximize his income, the owner should raise the rate nine times.
He will charge: .90 + 9·5 .= .$135 per night. As a result, he will lose 9·2 = 18 guests. . . He will have only: 72 - 18 .= .54 guests. But he will receive: .$135 × 54 .= .$7290 [Originally, his income was: .$90 × 72 .= .\$6450]

5. Ah, ok i understand now.

Thanks a lot to both of you!