Open Top Box
An open box with a square bottom is to be cut from a piece of cardboard 10 feet by 10 feet by cutting out the corners and folding the sides up.
Find the dimensions that will result in the largest volume.
Find the dimensions that will result in the largest volume.
Labels: outside-the-box
posted by Ragknot at
9:27 PM
16 Comments:
if the box is ten feet by ten feet, and what your saying is that we just cut the corners, then the box would be cut into a cross, with each part equaling 3.33 now to find the volume you would, go length x width x heigh = volume, so to find the volume of the box you just plug inthe 3.33 into our equation, which would be this
3.33 x 3.33 x 3.33 = 37.04 (which is rounded) so the box could hole 37.04 (inches, centimeters, or watever) squared of stuff
speed
To last post. wrong
Suppose you cut 3 foot squares from each corner. That leave 10 - 3 - 3 = 4 foot uncut. Folding up the uncut edges makes a box thats 4 foot by 4 foot by 3 foot high.
The bottom is 4 by 4 = 16 sqft. Three foot high makes 48 cuft.
That's easily bigger than your 37, but one could make it bigger than 48 with a little more planning.
the answer is 74.074
Sachin is right. The volume can be described by v=x*(10-2x)^2, where x is the side length of the squares cut out of the corners. Taking the derivative and setting that equal to zero yields x={5/3,5}. x=5 is obviously the x value that would yield minimum volume, so you should cut squares at each corner with side length 5/3. This yields a length of 20/3, a width of 20/3, and a height of 5/3, which yields a volume of 2000/27, or 74.074074074...
could u guys please explain how that x and 2x comes and that formula too??
4 squares need to be cut out of the sheet of cardboard so four sidescan be folded up to form an open-top box. They need to be squares and they all need to be the same size so that the four walls of the box are of the same height. Suppose we call the length of one of these squares, which is equal to the height of the box, x. Each side of the sheet of cardboard would have two of these squares cut out, hence 2x, leaving the side length to be 10-2x. The formula is just the volume formula for a rectangular prism, volume=length*width*height. As I said earlier, x is the height. The box has a square bottom, so the length and the width are equal. They are the remaining length of each side of the sheet of cardboard after the four squares have been cut out. Hence, the formula is v=x*(10-2x)*(10-2x), or v=x*(10-2x)^2.
how big do the corners have to be?... it only says the bottom has to be square
The corners cut out are 5/3 feet in length.
Cut nothing...just make the paper plain and raise up as les as u can 4 sides walls and at the corners fold the paper arround.If u make the walls tall u will have to fold more paper at the corners, so u lose volume.First off all is a logic problem and after can be veryfied mathematic.nop, don't cut nothing.
More than that, the problem didn't specify the size of the box, just the bottom to be square, so if i rise the walls just 1 inch i still have a box and folding the corners arround , will have 118*118*1 box
The method you're using by not cutting anything does not maximiz volume, only surface area.
To help folks visualize, here is a graph showing the volume of the box as a function of the size of cut corners. Note that the volume is zero, when nothing is cut or when 5 feet is cut (leaning nothing!) from each corner, and maximum when 1 2/3 ft. is cut.
The dimensions would be 10/3 feet by 10/3 feet by 10/3 feet because the one with the largest volume with the least surface area is a cube.
my maximized area multiply by 1 inch tall will still give the maximum volum.V=A*H....the area base is importand coz i loze minimum of cardbord.
you simply use calculus and find the derivative of the volume equation
james yu is wright...his volume will be 64000 cubbic inches, and mine one....13924 cubbic inches
Post a Comment
Links to this post:
Create a Link
<< Home