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Dimensions of box with largest volume

  1. Aug 27, 2008 #1
    1. The problem statement, all variables and given/known data
    Find the dimensions of a rectangular box with the largest volume with surface area 64cm^2.



    2. Relevant equations
    area of a rectangular prism 2xy+2yz+2xz


    3. The attempt at a solution
    took 2xy+2yz+2xz=64
    rearranging for z:
    z=(xy-32)/(y+x)
    partial derivative of z with respect to x
    y/(y+x)-(xy-32)/(y+x)^2
    partial derivative of z with respect to y
    x/(y+x) -(xy-32)/(y+x)^2
    setting both derivatives equal to zero to obtain the critical point....
    Here's where I hit a wall-
    I cannot get a real root for y.
    y/(y+x)-(xy-32)/(y+x)^2
    y^2+xy=xy-32
    y^2=-32
    Please only help me up to this point until I ask you to help me further, I
    want to challenge myself. Thank you!
     
  2. jcsd
  3. Aug 27, 2008 #2
    what are you maximizing?
     
  4. Aug 27, 2008 #3
    The volume of this box.
     
  5. Aug 27, 2008 #4
    you're maximizing xyz?
     
  6. Aug 27, 2008 #5
    Oh wait I'm finding the maximum dimensions that can be used to get the surface area of 64cm^2. So I maximize 2xy+2yz+2xz=64
     
  7. Aug 27, 2008 #6
    'maximum dimensions'?
     
  8. Aug 27, 2008 #7
    The problem states that I must find the dimensions of a rectangular box with the largest volume if it's total surface area is 64cm^2. What we don't know is how long those dimensions are.
     
  9. Aug 27, 2008 #8
    first you should find the maximum volume of a box with surface area 64cm^2.
     
  10. Aug 27, 2008 #9
    Okay! I'll take the problem from there and tell you how I made out.
     
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