SOLUTION: An open box, without a top is to be made by cutting a congruent squares from each corner of a rectangular sheet metal, which measures 5 inches by 8 inches, and folding up the side

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Question 1110914: An open box, without a top is to be made by cutting a congruent squares from each corner of a rectangular sheet metal, which measures 5 inches by 8 inches, and folding up the sides. Find the volume of the box, which has the greatest volume.
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
x, edge length of each square to remove
volume, %285-2x%29%288-2x%29x

x%2840-26x%2B4x%5E2%29
x%284x%5E2-26x%2B40%29
4x%5E3-26x%5E2%2B40x

dv%2Fdx=12x%5E2-52x%2B40

12x%5E2-52x%2B40=0
3x%5E2-13x%2B10=0
The root needed is ...

Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.
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

%28dV%29%2F%28dh%29 = 12h^2 - 52h + 40



and equate it to zero:

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

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

h%5B1%2C2%5D = %2813+%2B-+sqrt%2813%5E2+-4%2A3%2A10%29%29%2F%282%2A3%29 = %2813+%2B-+7%29%2F6.


There are two roots:  h%5B1%5D = %2813%2B7%29%2F6 = 20%2F6 = 10%2F3  and  h%5B2%5D = %2813-7%29%2F6 = 1.


Compare the values

    V((10/3) = 4%2A%2810%2F3%29%5E3+-26%2A%2810%2F3%29%5E2+%2B+40%2A%2810%2F3%29 = -7.4

and

    V(1)     = 4%2A1%5E3+-26%2A1%5E2+%2B+40%2A1 = 18.


The answer is  h = 1.

Solved.