A box (with no top) will be made by cutting squares of equal size out of the corners of a 38 inch by 46 inch rectangular piece of cardboard, then folding the side flaps up. Find the maximum volume of such a box. ROUND TO THE NEAREST CUBIC INCH.

Question

A box​ (with no​ top) will be made by cutting squares of equal size out of the corners of a 38 inch by 46 inch rectangular piece of​ cardboard, then folding the side flaps up. Find the maximum volume of such a box. ROUND TO THE NEAREST CUBIC INCH.

Reese King · Accepted Answer

Step 1Dimensions of cardboard   =  38  ×  46Since, squares of equal size are cut from all the sides. Valid length of size (say x) will be   &amp;gt;  0 and less than half of shorter side (i.e. 19), otherwise the new box will not have width. Thus, height of new box   =  x meh, length of new box   =  38  −  2  x, width of new box   =  46  −  2  x.Step 2Volume   =  (  46  −  2  x  )  (  38  −  2  x  )  ×  x  V  =  4      x    3    −  168      x    2    +  1748  xTo find maximum volume; derivative of volume should be put equal to 0 (to obtain roots of x)      v    ′    =  0  12      x    2    −  336  x  +  1748  =  0Roots   =  21.0946     and     6.9056.905 is smaller than 19  ∴  V  o  l  =  4      x    3    −  168      x    2    +  1748  x  =  4  (  6.905      )    3    −  168  (  6.905      )    2    +  1748  ×  6.905  ⇒  13386.835  −  8010.0762   ⇒  53767 cubic inch

A box (with no top) will be made by cutting squares of equal size out of the corners of a 38 inch by 46 inch rectangular piece of cardboard, then folding the side flaps up. Find the maximum volume of such a box. ROUND TO THE NEAREST CUBIC INCH.

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