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Math Help - Word problem involving Volume

  1. #1
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    Word problem involving Volume

    Hello

    I believe the answer to part a. of this question is 4 ballons. Part b. is beyond the scope of my abilities. Any help would be appreciated.

    a. If one Helium (gas) tank can fill 16 nine inch (diameter) spherical balloons, then the same tank can fill how many 3 feet (diameter) balloons?

    b. How many small marbles of radius r can be filled in a large cylindrical tumbler of height H and radius R ?

    Thanks,

    KP
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  2. #2
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    Hello, pashah!

    Your answer to part (a) is incorrect . . .
    . . and it seems to be a trick question (on a number of levels)


    a. If one Helium (gas) tank can fill 16 nine-inch diameter spherical balloons,
    then the same tank can fill how many 3-foot diameter balloons?

    A nine-inch diameter means a 4.5-inch radius.
    One nine-inch balloon holds: . \frac{4}{3}\pi(4.5^3) \:=\:121.5\pi in³ of helium.

    Since the tank can fill 16 of these balloons,
    . . the tank holds: . 16 \times 121.5\pi \:=\:1944\pi in³ of helium.


    Now a 3-foot diameter means a 36-inch diameter . . . an 18-inch radius.
    These larger balloons hold: . \frac{4}{3}\pi(18^3) \:=\:7776\pi in³ of helium.

    Hey! . . . the tank doesn't hold enough to fill even one 3-foot balloon.

    Answer: zero

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

    Oh, sure . . . you could fill one balloon part way.
    But then it wouldn't be a three-foot balloon, would it?

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  3. #3
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    Quote Originally Posted by pashah
    a. If one Helium (gas) tank can fill 16 nine inch (diameter) spherical balloons, then the same tank can fill how many 3 feet (diameter) balloons?
    If a helium tank can fill 16 nine inch diameter ballons then how much gas in the tank. Well, it is 16 times the amount of gase in each ballon. The amout of gas in a ballon is its volume. So you need to use the formula for volume,
    V=\frac{4}{3}\pi r^3. Since the diameter is 9 inches the radius is 4.5 inches. That means its volume is,
    \frac{4}{3}\pi (4.5)^3=121.5\pi. Therefore the total volume of the gas tank is that multiplied by 16,
    16\times 121.5\pi =1944 \pi cubic inches.

    Remember to keep using the same units. That means 36 inches diameter ballons. The volume of each one is, (radius is 18)
    \frac{4}{3}\pi (18)^3=7776 \pi
    Note that the amout of helium for one 3 foot one is beyond the amout of gas. So I think there is something wrong in your problem.
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  4. #4
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    Quote Originally Posted by pashah
    b. How many small marbles of radius r can be filled in a large cylindrical tumbler of height H and radius R ?
    This is a variant of the sphere packing problem. See here for example for a page devoted to this problem.

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