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October 20, 2014

October 20, 2014

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

Height:

Length:

Width:

- Applied Calculus -
**Reiny**, Friday, March 8, 2013 at 6:25pmbase = 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 6:28pmnew 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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