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Expert: - 7/19/2009


Question
Is it possible to work out the smallest surface area when you only have the following facts.
You need to find the smallest surface area for a suitcase when the dimensions are given as x + Y + z = 150. I know you can do it by trial and error ie substitution but is there an actual formula? Thanks

Answer
What gives the smallest surface area is a cube.
This would be a little bit difficult to carry.

To do it properly, solve for the sides of a rectangle to give the minimal edge.  If the size is the length l by the width w, the perimeter is 2w + 2l and the area is lw.  After doing some calculus, this would be found it needs to be square.

Next the volumn of a box given a square end would be solved for.
The box would have height h and edge on the base w (and also width).
This would, bu calculus, say the best volume would be when h=w.
The problem with this is it takes into no account the fact that the suitcase needs to be carried.

See, people would have a very difficult time carrying something cubic in shape, so the make them narrow.  Next, people have a hard time lifting it up in the air, so they aren't square.  They know what the average length is between an adults hand and the ground, and this must be the maximum height, but people very, so they probably make them as short as they can.

The only way to get volume to increase is to increase the length from front to back.

I'm not sure exactly how they determine what this will be, but there is a special number that makes rectangle look great.  If one of the sides is treated as 1 unit, the other side is made to be
(-1+rt(5))/2.  See, if one side has lenght a the height is b, then is a square is removed from one side the size is a-b x b.
If the ratio that was just given is used, a/b = b/(a-b).

A square can always be taken off the rectangle and the sides left will have the same ratio.  I watched a show on the perfect rectangle and there are numerous places in nature they appear.  It has been so long ago, I don't remember exactly where.  I do remember they occur in man made items and in natural items.

Now as far as the problem you gave me,
the largest volume would be if x = y = z = 50.
The volume would be 125,000 cubic units and
the surface area would be 15,000 sq units.

However, as was stated, this would be difficult to carry.

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