anonymous
  • anonymous
A closed box with square base is to be built to house an ant colony. The bottom of the box and all four sides are to be made of material costing 1 dollar/sq ft, and the top is to be constructed of glass costing 5 dollar/sq ft. What are the dimensions of the box of greatest volume that can be constructed for 72 dollars? NOTE: Let denote the length of the side of the base and denote the height of the box
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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anonymous
  • anonymous
i know how to do this but im having trouble coming to the correct answer
anonymous
  • anonymous
V = x^2 y ....... x = base we also know that 72 = 5(x^2) + 4(xy) + x^2 72 = 6x^2 + 4xy xy = (36 - 3x^2)/2 thus V = x(36 - 3x^2)/2 get the derivative and set it to zero, the resulting x-value is the length of the base of the box with the greatest volume.
anonymous
  • anonymous
can you help me do that cuz im apparently not doing it right

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anonymous
  • anonymous
Surface Area = 2 tops + 4 sides Area of square = 2 * x² + 4 * xy Total Cost: $5 *x² + $1* ( 4xy + x²) = $ 72 6 x² + 4xy = 72 =>Area: xy = (72 - 6x² ) / 4 = 18 - 1.5 x² => Volume: x ( xy) = 18x - 1.5 x³ Thus maximum Volume: V' = 18 - 4.5 x² = 0 => x = 2 ft, y = 6 ft

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