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Math Help - Calculus Word Problems

  1. #1
    Bar0n janvdl's Avatar
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    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
    -----------------------------

    1. Find the formula to calculate the income

    2. Calculate the maximum Income

    ------------------------------

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

    Thank You!
    Last edited by janvdl; May 14th 2007 at 04:51 AM. Reason: My bad grammar :-D
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  2. #2
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    earboth's Avatar
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    Quote Originally Posted by janvdl View Post
    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) = -10x + 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
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  3. #3
    Bar0n janvdl's Avatar
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    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...
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  4. #4
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    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 - 10x .(1)


    Maximize the income function: .I' = 0

    We have: .180 - 20x .= .0 . . . . x = 9 .**

    The maximum income is: .I .= .6480 + 1809 - 109 .= .$7290 .(2)

    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

    **

    To maximize his income, the owner should raise the rate nine times.
    He will charge: .90 + 95 .= .$135 per night.

    As a result, he will lose 92 = 18 guests.
    . . He will have only: 72 - 18 .= .54 guests.

    But he will receive: .$135 54 .= .$7290

    [Originally, his income was: .$90 72 .= .$6450]

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  5. #5
    Bar0n janvdl's Avatar
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    Ah, ok i understand now.

    Thanks a lot to both of you!
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