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Finding two possibilities of depth when volume is given?

  1. Mar 7, 2015 #1
    1. The problem statement, all variables and given/known data

    Problem - If the volume of the carry-on luggage is 2000 cubic inches, determine two possibilities for its depth. Where necessary, round to the nearest tenth of an inch

    2. Relevant equations

    V(x) = x(x+10)(30-2x)

    Volume function is given above

    3. The attempt at a solution

    I like to think I learn Math logic well, and I love Math...but this problem has me stumped...something is just not clicking in my brain when I try to start solving this problem.

    Now I set up the problem:

    2000 = x(x+10)(30-2x)

    I then said OK, I need to solve for x and by doing that I will find 1 solution, or maybe both, but I can't figure out how to combine the x's into one and then solve for the 1 remaining x.

    I tried finding the zeros of the 3 factors, but then wondered to myself why did I do that, it doesn't seem I need to do that.

    I was unable to ask my professor help on this problem because this was the last problem on my homework and I left the last couple of questions for today so I can bring in all my homework tomorrow (which is when it is due anyway) and I didn't think I would run into a mindwreck of a problem like this...not to mention this problem has a chance to be on the test tomorrow!!

    Would really appreciate help in solving this problem.
     
  2. jcsd
  3. Mar 7, 2015 #2

    LCKurtz

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    Apparently you aren't telling us everything you know about this problem. Anyway, simplify your equation and write it as a cubic equation = 0. Then look for a rational root to reduce it to a quadratic.
     
  4. Mar 7, 2015 #3
    The only thing I left out is the beginning format of the V(x) which was:

    (x)(x+10)(40-(x+x+10))

    The x being depth
    The x + 10 being length
    The (40 - (x+x+10)) being width(40) minus (depth + length)

    OK I compact the problem into the below:

    -2x3+30x2-20x2+300x

    Now I need a constant to try the Rational Zero Theorem

    So I simplify it as follows

    x(-2x2+10x+300)

    I then quickly jump to find a zero using -10 as a value for x to get zero

    I then use Synthetic Division which just brings me right back to the original:

    x(x+10)(-2x+30)

    And the roots from that is 0, -10, 15

    But I am not sure if finding the roots for this problem helps any...

    If it is any help, the 2 depth values is 7.8 inches and 10 inches that shows in the back of my text book as the solution...I just don't know how to get these 2 values?

    Should I be using another theorem or method to break this problem down?
     
  5. Mar 7, 2015 #4

    LCKurtz

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    You have to set the volume = 2000 and solve that cubic. You should get two positive values of ##x## which agree with your given answers.
     
  6. Mar 7, 2015 #5
    Another thing to note is this problem is at the end of a Chapter dealing with:

    The Rational Zero Theorm
    The Fundamental Theorem of Algebra
    The Linear Factorization Theorem
    Descartes Rule
     
  7. Mar 7, 2015 #6
    OMG unbelievable.

    Firstly, I didn't think about setting it equal to 2000 and then moving the 2000 over to make it equal to 0.

    Secondly, What was also messing me up was that I wasn't sure if looking for roots was the way to solve this, not realizing that was the way to solve it and that I had already found one of the answers (10 inches) when I found the zero for one of the simple factors from before.

    Thirdly, because I saw the problem asking for 2 possible depths, I was under the assumption that this problem would only result in 2 possible solutions, when in reality it was a couple solutions, some positive numbers and some negative numbers.

    Fourthly, it blew my mind how moving the 2000 over allowed me get a constant on the cubic equation...it totally transformed the entire problem as I started solving more and more of it, and I started to grin when I had to use the quadratic formula to get possible solutions with radicals, and one of those gave me the 7.8 inches (The second answer I was looking for)

    Thanks for the help!!
     
  8. Nov 13, 2015 #7
    Can anyone break this down with a step by step guide? After getting the roots, I'm not sure how to proceed.
     
  9. Nov 13, 2015 #8

    HallsofIvy

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    Well, what did you get for the roots? And what, exactly, does the problem ask you to find?
     
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