(On non-flushable paper)
From: Wesli AnneMarie Dymoke
To: Fellow Staff
Re: Toilet Paper
As cow-orkers, we appreciate each other's consideration in all aspects of our shared
experience, so let me first thank those who responded to my prior missive regarding
dental hygiene, including the few who apparently read my note about halitosis.
As we all desire greater respect in the workplace, I'm sure everyone will
appreciate the following.
While we all hope that everyone appreciates the importance of toilet paper,
not everyone may extend this beyond their own immediate needs. Not wishing to assume such inconsideration, it occurs
to me that some people may improperly estimate the amount of paper remaining on
a roll, and thus misjudge when it should be changed. Here, then, is a brief
primer on how to better estimate and know when to add a new roll, along with
some related pointers.
Let us begin with understanding how much paper is on a full roll. We can
avoid the complexity of axial-linear computation, understanding that the very
thin depth of the substrate makes this pointless, and instead calculate the
amount of paper in simpler terms of volume. We calculate this by comparing
cylinders of varying width. I will explain first how to calculate the whole
volume by overall diameter, and then from that how to estimate the proportional
volume of actual toilet paper by simple radius, or overall depth of paper from
the tube to the outside.
We calculate the volume of the cylindrical roll as: vr
= abhc where vr = the
volume of the roll, ab = the area of the
base, and hc = the height. (Understand that
"base" here refers to the round end of the cylinder, and "height" refers to the
length of the straight dimension perpendicular to the diameter of the round
"base". These terms are nominal, for mathematical convenience. Don't let the fact that
the toilet paper is mounted "sideways" confuse you.) This formula invokes a
component formula, the area of a circle (the round "base"), which we calculate
as: a = πr2 where a = area of the
circle (duh), π = the value of pi*, and r2 = the square of the radius of
the circle (base); thus the more verbose formula for the volume of a cylinder
is: vr = (πrb2)hc
. (*Engineers usually figure pi at
3.14159, but for our purposes we can just use the 3.14 they teach in public
schools. We're only talking about toilet paper, after all.)
The manufacturer supplies the (fixed) "height" of the roll as 4-½ in.;
meanwhile, using a very small tape measure that I carry with me for such
contingencies, I have measured the (full) diameter of the (unused) roll as
about the same (actually a bit less, but close enough for our purposes.) From
that, we first calculate the area of the "base" as: π(4.5/2)2
(4.5" diameter ¸ 2 = 2.25" radius.)
Thus, we calculate the area as 3.14 X ((4.5) X 2) to find an area of 63.585 square
inches. From that, we then calculate 63.585 X 4.5 to find an overall volume of 286.1325 cubic
inches. (I would normally prefer metric, but the convenient round imperial
figures may make these calculations clearer for some of you.) We can reasonably
round this figure to the simple integer of 286 cu. in.
Note the term "overall volume". Some portion of the total volume
of the roll does not include any usable hygienic paper, but instead is mostly
air and cardboard-the central tube or "core" of the roll. In order to know how
much paper is really in the roll, we must subtract the volume of the core, as: vp
= vr - vc where vp = volume of
paper, vr = the overall volume of the roll as we calculated
above, and vc = the volume of the mostly empty core.
I measured a core diameter of about 1-¾ in. (The cardboard tube itself
is just over one millimetre in thickness; I measured the outside diameter of
the tube, to better calculate actual volume of paper.) From that, I calculate
the volume of the core as π(1.75/2)2h = 10.81828125
cu. in., which we can safely round off to 10.8 cu. in.
Now, subtracting the volume of the core from the overall volume, we find
that 286 - 10.8 = 275.2 cu. in., a ratio of about 25.5:1; note that the
empty core takes up less than 4% of the total volume of the roll. Let's not get carried away.(In case any
of you are wondering, the total area of paper of the full roll is 225
square feet; I didn't calculate that-it's also printed on the packaging. A curious
person could figure out how much linear paper is left from calculating the
volume, but as I said, the paper is so thin as to make this almost purely
academic. Let's not get carried away.)
Now that we know the numbers, how can we use this knowledge to figure
out when a roll needs to be changed? Ideally, we would first decide how many
people (at least one) should be able to use it after us before changing, and
how much paper each person is likely to use, and figure backwards from there by
simple subtraction. There are two problems with this.
First, it's not really feasible to figure individual usage, even
as an average. To do so, we would have to find out how much paper people really
use, and the acquisition of that knowledge would be laborious, time-consuming,
and invasive. It's not really our business to know who is a folder or a
wadder-I doubt that most of us would want to volunteer such personal
information ourselves-or who is more fastidious about bathroom hygiene. There
are some things I don't need or want to know about any of you.
Second, there is an element of inconsideration in deciding for others
how much paper is enough. Some of us may not like others to make this kind of
decision for us. As a general rule of thumb, then, let us agree to estimate a
reasonable amount of remaining paper as the amount that you would use yourself,
multiplied by at least three. (In case this estimate may be inadequate or
disagreeable to some, I offer a stopgap measure, detailed afterwards.)
The most important thing is to be able to estimate remaining volume by
observing relative diameter. A roll that is about four and a half inches in
diameter is full, or nearly so. But how much is half a roll, paperwise? A
quarter roll of paper? Less?
First, don't forget about the empty core. The depth of paper around
the outside of the paper core is not 2-¼ inches, but only 1-¾ inches. So, a
little less than an inch (actually 7/8") depth means half
a roll left, right?
NO‼
Remember that volume does not function on straight linear measure. Let's do the math:
A paper depth of 1-¾ in. yields an overall
roll radius of 1.75 + .5 = 2.25 in. From that, figure that π2.252
= 15.9 square inches, so that 15.9 X 4.5 = 71.55 cubic inches. Subtract the core
volume, and we find that 71.55 - 10.8 = 60.75 cu. in., yielding a ratio
against the starting paper volume (275.2 cu. in., remember?) of 4.53:1- the
"half" roll is only 22% of the full volume!
At one quarter depth-7/8 in.-the variance is more
dramatic. 0.875 + .5 = 1.375 in. radius. Figure that π1.3752
= 5.9 square inches, \ 5.9 X 4.5 = 26.55 cubic
inches: a ratio of 10.37:1 against the full roll-the "quarter" roll is less
than one tenth of the full roll! That's less than sixty feet of paper left. (Okay, so
I went ahead and did the linear math, so what?) That may sound like a lot-about
twice the length of an average middle-income single-family home-but I also know
that some of you use way more than your fair share of paper. (Okay, so maybe I
do pay more attention to that than I let on earlier, so what?) That's less than 175
hockey tickets, at four inches long each.. (I got that figure off the package,
too, so I know it's correct.) Is it enough? How can you know for sure?
If you can't handle the math, then try this: Imagine you're home alone, and
the roll you see before you is the only roll of paper in the whole house, and
it must last you the next eight hours before you can get more. Are you
comfortable with how big the roll is? Will it last you the rest of the day?
Unless your answer is an emphatic YES, then in all moral fairness
you must change it, for the assured comfort and convenience of your
cow-orkers-including yourself, should you require a hasty revisit later, or in
case you're forced to work alongside someone else who might have been
inconvenienced by your or someone else's inconsideration.
Now, you may not be sure how you feel about the partially used roll.
Maybe it's just at that margin between comfortable and
uncomfortable. (In which case you cannot honestly supply an emphatic YES. Refer
to above.) Or maybe you're in a real hurry. (Please don't be in too much
of a hurry!) In this case, there is something else you can do, both to assuage
your own uncertain conscience, and to assure everyone's greater
happiness: Place a new, full roll within
reach of the toilet.
Now, be smart about this: The floor might be wet, or get wet (perhaps it
will be cleaned at some point?), in which case a roll placed on the floor near
the toilet might get wet. Likewise, raised surfaces near the toilet might offer
an opportunity for the new roll to be knocked off into the toilet, or
into the sink, or onto the floor. A wet sink (you do wash your hands, right?)
will ruin the paper as readily as a wet floor. The new roll also might not be
reachable for everyone. The back of the toilet is very close, certainly, but
reaching it might require a degree of flexibility that not everyone may
possess; how ironic that a roll you can lean back on is nevertheless out of
reach! Finally, the roll might not be noticed by everyone.
It's therefore best to place the new roll on top of the
partial one; just rest it carefully atop the old one, so that it's right there,
where no one can miss it, and anyone can reach it. Now, if you happen to find
the paper this way, don't just use the new roll and ignore the old one. That's lazy and stupid,
and also inconsiderate. What's the value of two partial rolls? Just carefully
remove the unused roll and set it aside (somewhere dry!), and use the partial
roll. If it runs out, grab the new one! (You did place it within your own
reach, and not chuck it across the room, right?)
If you change from an empty roll to a new one, don't just leave the
empty core on the spindle and prop the new roll on top of it. This is also lazy
and stupid, and also inconsiderate. Take the time to remove the empty core from
the spindle, and mount the new roll. I shouldn't have to explain how to do this. If you can
wipe yourself, you should be able to figure this out for yourself, too. (Though
in all honestly, I don't necessarily extend this much credit to every single
last one of you.)
Finally, if you use the last of the paper, and there is no more, don't just walk away
and leave it as a surprise for the next person. Close the lid (goddamn, I know
you can figure this out, guys, if you can follow the fucking play-offs-what the
hell's
wrong with you?) and place the empty core upright on the lid (so it won't roll off and
disappear, moron), so that the next person cannot miss this clear and obvious
signal that there is no more paper. If you're feeling especially generous, you might take
a detour on one of your frequent treks to the break room to bring some of the
paper from storage back to the bathroom. It's not too heavy for you, is it? I mean, it is
almost 4% air, right? And if there isn't any backup in storage, maybe you could divert
your overused mouth and abrasive voice to the practical goal of letting
other people know that there is no more toilet paper. Jesus Java-Chugging
Christ, would that really take too much out of your busy day?
