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Math Help - Solving the Volume of an Object with Polynomial Functions

  1. #1
    Newbie Xeaxy's Avatar
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    Solving the Volume of an Object with Polynomial Functions

    I've been stuck on a fairly difficult (well, for me at least) problem all day. It's concerning the volume in cubic feet of a CD holder:
    The volume in cubic feet of a CD holder can be expressed as
    V(x) = -x^3 - x^2 + 6x
    or, when factored, as the product of its three dimensions, depth, height, and width. The depth is expressed as (2 - x). Assume the height is greater than the width.
    The following are problems relating to the info above:
    a. Factor the polynomial to find linear expressions for the height and the width.

    b. Sketch a graph of the function. Find the x-intercepts. Explain what the x-intercepts represent geometrically.

    c. Describe a realistic domain for the function V(x) and explain why you consider it to be realistic.

    d. Find the maximum volume of the CD holder.
    I've been doing a lot of problems like this in my Algebra 2 class, and I just need a simple explanation on what to do. I know how to graph a polynomial function like this, so (b.) shouldn't be a problem.
    Last edited by Xeaxy; April 23rd 2010 at 02:06 PM.
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  2. #2
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    Quote Originally Posted by Xeaxy View Post
    The volume in cubic feet of a CD holder can be expressed as


    V(x) = -x^2 - x^2 + 6x

    or, when factored, as the product of its three dimensions, depth, height, and width. The depth is expressed as (2 - x). Assume the height is greater than the width.
    Are you sure the function is not -x^3-x^2+6x ??


    Quote Originally Posted by Xeaxy View Post
    a. Factor the polynomial to find linear expressions for the height and the width.
    Firstly take out x as a common factor, what are you left with?
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  3. #3
    Newbie Xeaxy's Avatar
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    Quote Originally Posted by pickslides View Post
    Are you sure the function is not -x^3-x^2+6x ??
    Oops, you're right. I fixed it.



    Quote Originally Posted by pickslides View Post
    Firstly take out x as a common factor, what are you left with?
    -x(x^2+x-6)

    I think.

    EDIT: I factored completely and it comes out to -x(x-2)(x+3). But I'm still confused as to how that represents the depth, height, and width of an object.
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    Quote Originally Posted by Xeaxy View Post
    EDIT: I factored completely and it comes out to -x(x-2)(x+3). But I'm still confused as to how that represents the depth, height, and width of an object.
    It does look confusing because you have what looks like a negative dimension in -x

    But they have given you a hint above in d =(2-x)

    so -x^3-x^2+6x = x(2-x)(x+3) ,x<2
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  5. #5
    Newbie Xeaxy's Avatar
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    Quote Originally Posted by pickslides View Post
    so -x^3-x^2+6x = x(2-x)(x+3) ,x<2
    Ok. So is x<2 the domain?

    Also, what do the x-intercepts represent geometrically?
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    Quote Originally Posted by Xeaxy View Post
    Ok. So is x<2 the domain?
    Not exactly x\in (0,2)


    Quote Originally Posted by Xeaxy View Post
    Ok. So is x<2
    Also, what do the x-intercepts represent geometrically?
    The other intercept x=-3 is no use to us, we don't want negative dimension.

    Look at a graph of the function. V(x) > 0,x \in (0,2)

    These 2 intercepts show where V(x)=0 we want to consider everything in between.
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  7. #7
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    Quote Originally Posted by pickslides View Post
    Not exactly x\in (0,2)




    The other intercept x=-3 is no use to us, we don't want negative dimension.

    Look at a graph of the function. V(x) > 0,x \in (0,2)

    These 2 intercepts show where V(x)=0 we want to consider everything in between.
    Ok, everything makes a lot more sense now. Thanks so much!
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