Dimensions of box with largest volume

  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. what are you maximizing?
     
  4. The volume of this box.
     
  5. you're maximizing xyz?
     
  6. 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. 'maximum dimensions'?
     
  8. 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. first you should find the maximum volume of a box with surface area 64cm^2.
     
  10. Okay! I'll take the problem from there and tell you how I made out.
     
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