How many square inches of material are needed to print 1,000 name tags?

Question

I don't understand this its says A hotel uses pentagonal-shaped name tags with demensions labeled below its a pentagonal shape with 3.5 in. and 2 in. and 3 in. and the answer choices are F 6,250 G 6,500 H 6,700 J 7,000 Could you please Take a look at this and give me some feedback please.

Answer 1

Hi Dinero W.,

By no means may you just divide by the pentagon area to obtain required material area! First, as Philip P. points out, you have not supplied enough information to make your pentagon. Second, there is no close-fitting plane tiling in pentagonal tiles, only pseudo-symmetry (you can find out more on that, if you Google thoughtfully). Third, may you "flip" the pentagon specified (if the edges are given in a specific rotational direction), i.e.turn it over for arranging and cutting (that will require printing on both sides!)? You may find that alternate pentagons (reading "across" a row), fit closest if alternate ones are rotated 180 degrees or so. Fourth, you must account for the unused areas at all borders. That process will require you to know the lateral dimension of the material supplied, if it comes as a roll.

Now, you might want to do some modelling on your own, using identical sized circles to fill a deep rectangular "box" planar area, as a function of the ratio of the box width to the diameter of the circles. There is a tradeoff between stacking in columns, vs. stacking in a hexagonal array (for the circle centers), which is sensitive to the ratio and the depth of the box. (You pack more densely up and down, but you lose one circle on alternate rows, for hexagonal.) The most-efficient filling flips back and forth between those two "modes", as the box gets "larger", until the box gets "deep" enough and "wide" enough, then the hexagonal is always better. It's a neat little problem, but within high-school geometry skills (since angles, etc. are all either 90 degrees or 60 degrees, variously). Oh yes, not only do you have to optimize the bulk packing, but you must consider the top row, which may be subject to irregular constraints!

-- Cheers, --Mr. d.