Question 1110914
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Cutting the (h x h)-squares from the 5X8 inches metal sheet and folding, you get the rectangular prism (open box) of dimensions

    (5-2h) x (8-2h) x h


with the volume of   V(h)= h*(5-2h)*(8-2h) = 4h^3 -26h^2 + 40h cubic inches.



To find the maximal volume, take the derivative

{{{(dV)/(dh)}}} = 12h^2 - 52h + 40



and equate it to zero:

12h^2 - 52h + 40 = 0,

3h^2 - 13h + 10 = 0,

{{{h[1,2]}}} = {{{(13 +- sqrt(13^2 -4*3*10))/(2*3)}}} = {{{(13 +- 7)/6}}}.


There are two roots:  {{{h[1]}}} = {{{(13+7)/6}}} = {{{20/6}}} = {{{10/3}}}  and  {{{h[2]}}} = {{{(13-7)/6}}} = 1.


Compare the values

    V((10/3) = {{{4*(10/3)^3 -26*(10/3)^2 + 40*(10/3)}}} = -7.4

and

    V(1)     = {{{4*1^3 -26*1^2 + 40*1}}} = 18.


The answer is  h = 1.
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Solved.