chris215
  • chris215
A box is to be constructed from a sheet of cardboard that is 10 cm by 50 cm by cutting out squares of length x by x from each corner and bending up the sides. What is the maximum volume this box could have? (Round your answer to two decimal places.)
Mathematics
chestercat
  • chestercat
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chris215
  • chris215
i got 2.36 is that right?
merchandize
  • merchandize
yes
merchandize
  • merchandize
So the box will have height x, length 10 - 2x, and width 50 - 2x, so we have: v = x(10 - 2x)(50 - 2x) v = x(500 - 20x - 100x + 4x^2) v = 4x^3 - 120x^2 + 500x dv/dx = 12x^2 - 240x + 500 Set that to zero: 12x^2 - 240x + 500 = 0 x = (-(-240) +/- sqrt((-240)^2 - 4*12*500)) / (2*12) x = (240 +/- sqrt(57600 - 24000)) / 24 x = (240 +/- sqrt(33600)) / 24 x = 10 +/- sqrt(175/3) But if x is greater than 5, the length of the box is negative, which doesn't make sense, so x can't be 10 + sqrt(175/3), which is approximately 17.6, so therefore x = 10 - sqrt(175/3) =~ 2.36

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