Question

A square piece of cardboard 1 foot on each side is to be cut and folded into a box without the top(small squares will be cut off from each corner). Find the dimensions of the box with the greatest volume

Answer 1

okay. suppose x feet are cut from each side. each of the cut outs will have an area of x^2.
that means the height is x feet.
let the length be L and the width be W
L = 1-2x
W = 1-2x
volume is LWH
V = x(1-2x)^2
to find the greatest volume, find out dV/dx and set it to 0

Answer 2

the heigh (h) would be x??

Answer 3

yes, try to visualize it. take a flat sheet of paper, cut out square shapes from each edge and try making it into a open box shape

Answer 4

i get the derivative as being : 1-8x+12x^2

Answer 5

okay! I havent done it, I can only tell you the method. I trust you can take it from there.

Answer 6

so, i got the derivative as being dv/dx=1-8x+12x^2, then i got its roots as being 1/6 and 1/5. How do i find out what dimensions maximize the value?

Answer 7

substitute each in the volume and see which one gives a bigger volume

Answer 8

can you check if what i did is right, cause i got fractions as the critical points

Answer 9

yes, you should get fractions. the side of the cardboard is 1 foot.
so x is less than 1 foot, so it is logical that your answers should be less than 1.

Answer 10

okay so i pliugged those citical values: 1/5 and 1/6 into the orginal volume formula: v=(1-2x)^2*x. For 1/5 i got .072. For 1/6 i got .074. Does this mean that 1/6 will gets me the greats volume, which also means that the dimension of the box should be 1+1/6 by 1+1/6 ???

Answer 11

yes, then 1/6 is the correct value for x
the dimensions of the cardboard are 1 foot by 1 foot. why would it be 1+1/6 by 1+1/6?

Answer 12

Well, the question as me to find the dimensions of the box with the greatest volume?

Answer 13

so the dimensions are L,W and H
which are 1-2x,1-2x and x

Answer 14

but where does the 1/6 come inot play, by that i mean what does it reperesent?

Answer 15

okay. do exactly as I say. take a sheet of paper.
tell me when you have done that.

Answer 16

okay i ahve a sheet of paper

Answer 17

okay, cut it so that it is a square sheet of paper if it is a rectangular sheet.
if it is already a square sheet of paper, leave it as it is.

Answer 18

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http://www.twiddla.com/573702