Posted by **Jacob** on Friday, March 8, 2013 at 4:55pm.

If an open box is made from a tin sheet 7 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.)

Height:

Length:

Width:

- Applied Calculus -
**Reiny**, Friday, March 8, 2013 at 6:25pm
base = 7- 2x

height = x

volume = x(7-2x)^2

= 49x - 28x^2 + 4x^3

d(volume)/dx = 49 - 56x + 12x^2

= 0 for a max volume

12x^2 - 56x + 49 = 0

x = (56 ± √784)/24

= (56 ± 28)/24 = 3.5 or 7/6 or 1.1666...

but clearly x < 3.5 or we have cut the whole base away.

base is 7 - 2(7/6) = 14/3 by 14/3

and the height is 7/6

round to your required decimals

- Applied Calculus -
**Damon**, Friday, March 8, 2013 at 6:28pm
new length = 7 - 2 h

height = h

volume = (7-2h)(7-2h)(h)

v = (49 -28 h + 4 h^2)h

so

v = 4 h^3 -28 h^2 + 49 h

dv/dh = 0 for max or min

dv/dh = 12 h^2 -56 h + 49 = 0

(6h -7)(2 h -7) = 0

h = 7/6 or h = 7/2

if h = 7/2, the box has zero bottom

so the answer is h = 7/6

7 - 2(7/6) = 7-7/3 = 14/3 = length and width

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