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Homework Help: Hot water cylinder

  1. May 28, 2010 #1
    1. The problem statement, all variables and given/known data

    To reduce heat loss, the surface area of a hot-water tank must be kept to a minimum. If such a tank is 125 litters in capacity, and can be approximated by a cylinder in shape with a hemispherical end cap; calculate the radius and overall height for minimum heat loss.


    hi can anyone check my ansers please thanks


    3. The attempt at a solution
    125 litres = 125000 cm^3

    The volume is the sum of the volume of a hemisphere and a cylinder.
    V= 2/3 πr^3+πr^2 h
    The surface area is
    S=2πrh+2πr^2
    Isolate h in the Volume equation.
    V= πr^3+πr^2 h

    h=(V- 2/3 πr^2 )/(πr^2 )

    Substitute for h into the Surface area equation.
    S=2πr^2 (V- 2/3 πr^2 )/(πr^2 )+2πr^2
    S=(2V )/r+2/3 πr^2



    calculate the derivative.
    ds/dr= - (2V )/r^2 +4/3 πr
    Solve for r
    - 2V + 4/3 πr^3=0

    r^3= (3v )/2π

    r^3= (3 ×125000 )/2π

    r^3= 59683
    r= ∛59683
    r = 39.1 cm
    to calculate the height
    h=(125000- 2/3 π〖39.1〗^2 )/(π〖39.1〗^2 )

    Height = 25 cm


    sorry can figuare how to make the equations non linnear
     
  2. jcsd
  3. May 28, 2010 #2
    Your answer looks right to me; I didn't check the last couple calculations, but i get r = 39.1 cm, too.
     
  4. May 28, 2010 #3
    thanks did i include the base ok?
     
  5. May 28, 2010 #4
    Oh wait, I might have duplicated your mistake. :) The cylinder just has a hemispherical cap in place of one of the ends, but the other end is just a flat "circle" cap, right?

    Then:

    [tex] SA = SA_{walls} + SA_{flatcap} + SA_{hemcap} = 2 \pi r h + \pi r^2 + 2 \pi r^2 = 2 \pi r h + 3 \pi r^2 [/tex]

    And then solve in the same way you did.
     
  6. May 28, 2010 #5
    thank s
     
  7. May 29, 2010 #6
    do i use the 2pirh + 3pir^2 for making h the subject

    am hopeless at this question
     
  8. Jun 2, 2010 #7
    hi i was check over my height equation is it

    2pir^2 (v- 0.66 pi r^3/pi r^2) = height?
     
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