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Metal Fabrication If an open box is made from a tin sheet 10 in. square by cutting out identical squares from each corner and bending up the resulting flaps, determine the dimensions of the largest box that can be made. (Round your answers to two decimal places.)

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Answer:

  • ength (l) : (10-2*5/3) = 20/3
  • width(w): (10 - 2*5/3) = 20/3
  • height(h): 5/3

Step-by-step explanation:

Let x is the side of identical squares

By cutting out identical squares from each corner and bending up the resulting flaps, the dimension are:

  • length (l) : (10-2x)
  • width(w): (10-2x)
  • height(h): x

The volume will be:

V = (10-2x) (10-2x) x

<=> V = (10x-2[tex]x^{2}[/tex]) (10-2x)

<=> V = 100x -20[tex]x^{2}[/tex] - 20[tex]x^{2}[/tex] + 4[tex]x^{3}[/tex]

<=> V = 4[tex]x^{3}[/tex]  - 40[tex]x^{2}[/tex] + 100x

To determine the dimensions of the largest box that can be made, we need to use the derivative and and set it to zero for the maximum volume

dV/dx = 12[tex]x^{2}[/tex] -80x + 100

<=>  12[tex]x^{2}[/tex] -80x + 100 =0

<=> x = 5 or x= 5/3  

You know 'x' cannot be 5 , because if  we cut 5 inch squares out of the original square, the length and the width will be 0. So we take x = 5/3

=>

  • length (l) : (10-2*5/3) = 20/3
  • width(w): (10 - 2*5/3) = 20/3
  • height(h): 5/3

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