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Math HW

  1. Aug 20, 2008 #1
    1. The problem statement, all variables and given/known data

    1. A small resort in Payatas expects 30 guests if they charge P1000 per room. However, they notice that for every P50 increase in this rate, the number of expected guests decreases by one. If the company shoulders P100 as operational cost for every room that they lease,
    (a) how much revenue would the resort get if they accommodate 28 guests?
    (b) how much profit would the resort gain if they charge P1400 per room?
    (c) what is the ideal number of guests? (i.e. that maximizes the profit)
    (d) how much should the resort charge the use of its rooms to meet the ideal number of
    guests?

    2. The equation
    x^2 /16 + y^2 /9 = 1
    is an ellipse with center at the origin and
    whose graph appears on the right. Find the
    area of the square that can be inscribed (or
    fitted) in this ellipse.

    3. Suppose the function f(x) has 16 distinct zeros whose sum is 23, find the sum of the zeros of
    the function f(3x − 4).

    2. Relevant equations

    i dont know how to get the quadratic eqn for 1.


    2. x^2 /16 + y^2 /9 = 1

    x = y therefore x^2 /16 + x^2 /9 = 1
    x = 12/5 and -12/5

    what does that mean?

    3. sorry i have no clue how to answer this
     
  2. jcsd
  3. Aug 20, 2008 #2

    HallsofIvy

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    This is ambiguous as they might get 28 guests no matter how much they charge. However, you are supposed to assume that this 28 guests fits the "expected" number. 28 is 2 less than 30 so how much more than 1000 were they charging each guest? 28 guests at that charge per guest is how much?

    Much the same. 1400 is 400= 8(50) so how many guests should they expect?

    For these last two you will need the general formula which you should be able to get by thinking about (a) and (b). If you want C= f(N), how much they charge as a function of number of guests, start with 1000 and subtract 50 for each guest less than 30 (i.e. subtract 50(30- N). Of course, the revenue is the charge times the number of guests. Since that is a quadratic function you find the maximum (vertex of the parabola) by completing the square.


    It doesn't mean anything. You have no reason to think that "y= x" on the ellipse. What are the lengths of the two axes? Have you drawn a picture? Do so and then draw a square around it. Notice that it must fit the LONGEST axis.

    3. sorry i have no clue how to answer this[/QUOTE]
    Saying that a polynomial has "16 distinct zeros" means it can be written as a product:
    [itex](x- a_1)(x- a_2)(x- a_3)\cdot\cdot\cdot(x- a_{16}[/itex]
    That their sum is 23 tells you that [itex]a_1+ a_2+ a_3+ \cdot\cdot\cdot+ a_{16}= 23[/itex].
     
    Last edited: Aug 20, 2008
  4. Aug 20, 2008 #3
    a. each room is 1100

    28 guests * 1100 = 30800
    but i should subtract 2800 because 100 is subsidized by the 'resort

    a. should b **28000**

    b. at 1400 there are 22 guests...
    1400*22 = 30800
    subtract 2200

    b. should be **28600**

    how should i get the quadratic eqn to solve for the max. pt?
     
  5. Aug 20, 2008 #4

    tiny-tim

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    Hi hancyu! :smile:

    I think they mean the largest "upright" square, with its sides parallel to the axes.

    EDIT: I should have drawn a diagram first :redface:

    always draw a diagram!

    No, I don't think they meant that … but the result is the same …

    at least one diagonal of the square must touch the ellipse at both ends …

    so at what angle is the shorter of two perpendicular chords of the ellipse the longest? :wink:

    Then the points at which the chord touches the ellipse will be on the line … ? :smile:
    (it needn't be a polynomial)

    If you're confused :confused:, go one step at a time :wink:

    If the zeros of f(x) are a b c d …,

    what are the zeros of f(3x)?

    And then what are the zeros of f(3x - 4)? :smile:
     
    Last edited: Aug 20, 2008
  6. Aug 20, 2008 #5
    the zeros of f(3x) are 3x-a, 3x-b, 3x-c, 3x-d...

    then the zeros of f(3x-4) are 3x-4a, 3x-4b...

    is that correct?
     
  7. Aug 20, 2008 #6

    tiny-tim

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    No … if 3x-a was a zero of f(3x), then f(3(3x-a)) would be zero … and it isn't.

    Hint: f(a) = 0, so f(3(what)) = 0 ? :smile:
     
  8. Aug 20, 2008 #7
    a/3 ?

    so the zero should be..... a/3 +4 , b/3 +4...
     
  9. Aug 20, 2008 #8

    tiny-tim

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    erm … is f(3(a/3 + 4) - 4) = 0? :rolleyes:

    (always write things out in full!)
     
  10. Aug 20, 2008 #9
    f(a/3 + 4/3) = 0 ?
     
  11. Aug 20, 2008 #10

    HallsofIvy

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    I would say that the revenue is 30800 while the profit is 30800- 2600

    I pretty much gave you the formula for the price that will produce a given number of guests: "start with 1000 and subtract 50 for each guest less than 30 (i.e. subtract 50(30- N)."
    That, you can expect N guests if you charge C= 1000- 50(30-N)= 1000- 1500+ 50N= 50N- 500. Revenue then is CN= (50N- 500)N= 50N2- 500N. The profit is given by that minus cost (100N) so profit is 50N2- 500N- 100N= 50N2- 600N.
     
  12. Aug 20, 2008 #11

    tiny-tim

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    :biggrin: Woohoo! :biggrin:

    (I'm assuming you mean f(3(a/3 + 4/3) - 4) = 0 :wink:)

    So the sum of all 16 zeros of f(3x - 4) is … ? :smile:

    (btw, how are you doing with the ellipse/square question)
     
  13. Aug 20, 2008 #12
    how do i get the sum...there are 16 unknown variables...a,b,c,d,e...p


    i think i got the answer on the ellipse problem.... length of each side is 24/5

    area is 576/25.
     
  14. Aug 20, 2008 #13

    tiny-tim

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    That's it! :smile:
    Hint: call the new zeros a',b',c',d',e'...p'.

    What is the equation that gives you the value of a'? :smile:
     
  15. Aug 20, 2008 #14
    f(3(a/3 + 4/3) - 4) = 0 ?
     
  16. Aug 20, 2008 #15

    tiny-tim

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    Yes, but with a' on the LHS:

    a' = a/3 + 4/3. :rolleyes:

    So what is a' + b' + … + p' ? :smile:
     
  17. Aug 20, 2008 #16
    im sorry. i really dont get this. whats a' ?
     
  18. Aug 20, 2008 #17

    tiny-tim

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    The zero of f(3x-4) corresponding to a. :smile:
     
  19. Aug 20, 2008 #18
    what is a' + b' + … + p' ? how do i add it? there are many different variables?

    is it 1/3(a....p) + 64/3 ?
     
  20. Aug 20, 2008 #19

    tiny-tim

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    Yes of course it is! :smile:

    (Why all the questions? :confused:)

    Now can you write it all out properly (perhaps using a1 a2 … a16 instead of a b … p)? :smile:
     
  21. Aug 20, 2008 #20
    ok...

    1/3(a1+a2+a3+a4+a5+a6+a7+a8+a9+a10+a11+a12+a13+a14+a15+a16) + 64/3 = 0

    so... the sum is 64?
     
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