Chocolate Box Company is going to make open-topped boxes out of 9 × 14-inch rectangles of cardboard by cutting squares out of the corners and folding up the sides. What is the largest volume box it can make this way?

Question

amistre64 · Answer

9-2x = width
14-2x = length
x = height

Area of a box is: WLH

A = (9-2x)(14-2x)(x)
A = (9-2x)(14x-2x^2)
A = 126x -18x^2 - 28x^2 +4x^3
A = 4x^3 -46x^2 +126x

get the derivative and solve it for 0:

Dx(A) = 12x^2 -92x +126 = 0

This would be easier if I had pencil and paper handy :)

amistre64 · Answer

the answer will have to fall between 0 and 9, so we can throw out anything that is out of those bounds.

amistre64 · Answer

x^2 - (92/12)x +(92/24)^2 = -(126/12) + (92/24)^2
(x - (92/24))^2 = -(6048/576) + (8464/576)
x - 92/24 = +-sqrt(2416)/24
x = +-sqrt(2416)/24 + 92/24

x = 5.881  or x= 1.785

remember that x is doubled; so 5.88 doubled is over 9

I would say that x would be about 1.785  :=: 2

A = (9-2(2)) (14-2(2))(2)
A = (5)(10)(2)
A = 25 cubic inches if I did it right :)

amistre64 · Answer

And of course I would forget how to multiply after all that ... lol

5(10) = 50;   50(2) = 100

A equals about 100 cubic inches....

amistre64 · Answer

101.09329869793039430976432210189  if you want to be more accurate :)

anonymous · Answer

THANK YOU!!! Amistre64 it was right...your a life saver.

anonymous · Answer

amistre64 is correct