Results 1 to 7 of 7

Thread: Find the value of x such that the volume is a maximum

  1. #1
    Newbie
    Joined
    Mar 2017
    From
    UK
    Posts
    8

    Find the value of x such that the volume is a maximum

    Morning all,

    Not sure if i'm making a double post here not, but got an assignment back, and seems i've gone wrong on just one of the answers. Therefore, I was hoping someone could help me out

    Gut feeling, is that i'm approaching the question wrong, but we will see:

    Question:

    From a rectangular sheet of metal measuring 120 mm by 75 mm equal squares of side x are cut from each of the corners. The remaining flaps are then folded upwards to form an open box. Show that the volume of the box is given by:
    V = 9000x 390x2 + 4x3
    Find the value of x such that the volume is a maximum.

    Answer:
    I wasn't entirely sure how to approach it, and ended up with something like this (see image):

    Find the value of  x  such that the volume is a maximum-capture.png

    Thanks in advance for any help
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    15,765
    Thanks
    3492

    Re: Find the value of x such that the volume is a maximum

    $V=H \cdot L \cdot W$

    $V=x(120-2x)(75-2x)$

    $V=(120x-2x^2)(75-2x)$

    $V=120x(75) + 120x(-2x) -2x^2(75) -2x^2(-2x)$

    $V=9000x-240x^2-150x^2+4x^3$

    $V=9000x-390x^2+4x^3$

    So ... since you posted your problem in the geometry forum, I have to ask what method were you to use to determine the maximum volume?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Mar 2017
    From
    UK
    Posts
    8

    Re: Find the value of x such that the volume is a maximum

    That makes a lot more sense, thank you - and to my embarrassment this wasn't supposed to end up in the geometry section lol
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    15,765
    Thanks
    3492

    Re: Find the value of x such that the volume is a maximum

    Did you determine the value of $x$ that yields a maximum volume?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Mar 2017
    From
    UK
    Posts
    8

    Re: Find the value of x such that the volume is a maximum

    Quote Originally Posted by skeeter View Post
    Did you determine the value of $x$ that yields a maximum volume?
    I'll be honest, i'm not entirely sure what the question is asking me to find, its the "that yields a maximum volume" that's throwing me.

    Do i effectively just turn it into a quadratic equation after finding the derivative?
    Last edited by barnsey471; Mar 20th 2017 at 08:11 AM.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    15,765
    Thanks
    3492

    Re: Find the value of x such that the volume is a maximum

    Note that $0 < x < \dfrac{75}{2}$ (why?) ... some value of $x$ in that interval will result in a box with the largest volume.

    One can use calculus to determine the value of $x$ where volume is a maximum ...

    $V = 9000x-390x^2+4x^3$

    $\dfrac{dV}{dx} = 9000 - 780x + 12x^2$

    set $\dfrac{dV}{dx} = 0$ ...

    $12x^2 - 780x + 9000 = 0$

    $12(x^2-65x+750) = 0$

    $12(x-15)(x-50) = 0$

    mathematically, either $x=15$ or $x=75$ ... but recall the restriction mentioned earlier for the value of $x$. This says $x=15$ is a candidate for yielding a max volume.

    Using the 2nd derivative test ... $\dfrac{d^2V}{dx^2} = 12(2x-65) \bigg|_{x=15} < 0 \implies V(15)$ is a maximum.

    $V(15) = 15[120-2(15)][75-2(15)] = 15 \cdot 90 \cdot 45 = 60750 \, mm^3$



    If all this is unfamiliar, one may use technology (a grapher) to determine maximum volume by looking at the graph of $V = 9000x-390x^2+4x^3$ ... see attached
    Attached Thumbnails Attached Thumbnails Find the value of  x  such that the volume is a maximum-max_vol.png  
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor
    Joined
    Feb 2014
    From
    United States
    Posts
    1,703
    Thanks
    802

    Re: Find the value of x such that the volume is a maximum

    Quote Originally Posted by barnsey471 View Post
    I'll be honest, i'm not entirely sure what the question is asking me to find, its the "that yields a maximum volume" that's throwing me.

    Do i effectively just turn it into a quadratic equation after finding the derivative?
    Please see skeeter's answer, which goes into detail.

    The derivative of a cubic function is a quadratic function.

    In this case, $V = 9000x - 390x^2 + 4x^3 \implies \dfrac{dV}{dx} = 9000 - 780x + 12x^2.$

    A minimum or maximum requires that $\dfrac{dV}{dx} = 0.$

    So $\dfrac{dV}{dx} = 0 \implies 9000 - 780x + 12x^2 = 0,$ which is a quadratic equation.

    It helps to keep your vocabulary straight.

    When finding extrema of a differentiable function, you calculate the derivative of the function, which is a new function, set that new function equal to zero, and solve. You then must test your solutions to find which are maxima, minima, and points of inflection. That is the basic practical use of differential calculus.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: Nov 15th 2012, 08:45 PM
  2. Replies: 5
    Last Post: Nov 1st 2012, 10:56 PM
  3. Maximum possible volume
    Posted in the Calculus Forum
    Replies: 7
    Last Post: Jun 10th 2011, 09:32 AM
  4. Find Maximum Volume
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Apr 18th 2010, 08:49 AM
  5. how to find maximum volume of cube inside cone
    Posted in the Calculus Forum
    Replies: 8
    Last Post: Jun 13th 2009, 11:07 PM

/mathhelpforum @mathhelpforum