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Thread: Cuboid tank help

  1. #1
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    Cuboid tank help

    A cuboid tank is open at the top and the internal dimensions of its base are x m and 2x m.
    The height is h m.
    The volume of the tank is V cubic metres and the volume is fixed. Let S mē denote the internal surface area of the tank.

    S in terms of x and h
    $\displaystyle S=2(2xh)+2x(x)+2(hx)$
    $\displaystyle S=2x^2+6xh$

    S in terms of V and x
    $\displaystyle V=2x * x * h $

    $\displaystyle V=2x^2h$

    $\displaystyle h=\frac{V}{2x^2}$

    $\displaystyle S=2x^2+6xh$

    Sub $\displaystyle h=\frac{V}{2x^2}$

    $\displaystyle S=2x^2+6x\left(\frac{V}{2x^2}\right)$

    $\displaystyle S=2x^2+\frac{3V}{x}$

    The maximal domain of S in terms of V and x is $\displaystyle (0,\infty)$

    If $\displaystyle 2<x<15$ find the maximum value of S if $\displaystyle V = 1000m^3$

    I'm stuck here, can anyone please help?

    Here's a visual representation of the cuboid
    Attached Thumbnails Attached Thumbnails Cuboid tank help-cuboid.png  
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  2. #2
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    Quote Originally Posted by user_5 View Post
    A cuboid tank is open at the top and the internal dimensions of its base are x m and 2x m.
    The height is h m.
    The volume of the tank is V cubic metres and the volume is fixed. Let S mē denote the internal surface area of the tank.

    S in terms of x and h
    $\displaystyle S=2(2xh)+2x(x)+2(hx)$
    $\displaystyle S=2x^2+6xh$

    S in terms of V and x
    $\displaystyle V=2x * x * h $

    $\displaystyle V=2x^2h$

    $\displaystyle h=\frac{V}{2x^2}$

    $\displaystyle S=2x^2+6xh$

    Sub $\displaystyle h=\frac{V}{2x^2}$

    $\displaystyle S=2x^2+6x\left(\frac{V}{2x^2}\right)$

    $\displaystyle S=2x^2+\frac{3V}{x}$

    The maximal domain of S in terms of V and x is $\displaystyle (0,\infty)$

    If $\displaystyle 2<x<15$ find the maximum value of S if $\displaystyle V = 1000m^3$

    I'm stuck here, can anyone please help?
    using calculus, find $\displaystyle \frac{dS}{dx}$ , determine the critical value and confirm that it is a value of x that minimizes S

    using technology, graph S(x) in the window 2 < x < 15 and locate the minimum.
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