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Math Help - Height and radius of cylinder with the least possible surface area given the volume

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    Height and radius of cylinder with the least possible surface area given the volume

    A cylinder has a volume of 100 cubic inches. What is the height and radius of such a cylinder with the least possible surface area?

    Constant volume of a cylinder = (pi)r^(2)h

    Surface area, including both ends = 2(pi)r^2 + 2(pi)rh
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  2. #2
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    Re: Height and radius of cylinder with the least possible surface area given the volu

    Therefore h = \frac{100}{\pi r^2} now minimise surface area in terms of height.
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    Re: Height and radius of cylinder with the least possible surface area given the volu

    Quote Originally Posted by Roleparadise View Post
    A cylinder has a volume of 100 cubic inches. What is the height and radius of such a cylinder with the least possible surface area?

    Constant volume of a cylinder = (pi)r^(2)h

    Surface area, including both ends = 2(pi)r^2 + 2(pi)rh
    (pi)r^2h=100 So h=100/(pi)r^2 Sub this into S getting S=2(pi)r^2+200/r Put derivative=0 to find required r and then h
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    Re: Height and radius of cylinder with the least possible surface area given the volu

    You can let h = \frac{100}{\pi r^2} and substitute h into your surface area function and differentiate with respect to r.


    Or, if you know how to use Lagrange multipliers, you can use them: there exists a constant \lambda such that


    \nabla A = \lambda \nabla V, where A(r,h) and V(r,h) are the surface area and volume functions, respectively.
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