1 The Filmy Dome Spider The following photograph of the web of a

The Filmy Dome Spider
The following photograph of the web of a filmy dome spider was taken by a
mathematician. You can find information and photographs about filmy dome spiders at
the following websites:
http://www.pestproducts.com/spider-webs.htm
http://hartmanprehistoricgarden.com/sa-neriene.html
http://www.biokids.umich.edu/guides/tracks_and_sign/build/webs/
http://bugguide.net/node/view/323460/bgimage
http://www.sergeev.us/pictures/archives/compress/2009/721/24s.htm
The mathematician claims that the cross-section of the spider web in this photo is a
perfect parabola. Use this photograph to build a mathematical model that can demonstrate
how accurate this claim is. Show all your work. You may use computer applications for
this task.
PLEASE USE A BLACK PEN, ROLLERBALL, OR FELTTIP WRITER AND
ANSWER THE QUESTIONS ABOVE ON A SEPARATE PIECE OF PAPER
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What this tasks accomplishes
Filmy dome spiders appear to make webs in the form of an upside down parabolic cup
(paraboloid). Students are presented with a photograph of such a web and are asked to
determine the quadratic function that describes the shape of the cross section of this web.
They must also determine how well (correlation) this model fits the curve. They may use
dynamic software, such as Geogebra or Geometer’s Sketchpad for this task.
(Mathematical modeling, Quadratic functions, standard form, vertex form, symmetry,
curve of best fit, correlation coefficient.)
Possible Solutions
This is a photo I took of a filmy dome spider web that was taken head on. This is actually
a 3-D web that looks like an upside down parabolic cup. I was curious to see if the shape
was actually parabolic. Most spider silk will form catenaries, but this inverted shape
appeared very parabolic to me. (The light blue dot is a speck of dirt caught in the web.)
In the picture below you can see that I was able to fit a curve reasonably well. I imported
the picture to Geometer’s SketchPad to plot the points (measure the coordinates) and
used some trial and error for the curve after I put in one point value for y=ax2. I put the
vertex of the “parabola” in (0,0).
4
2
-15
-10
-5
5
10
15
20
-2
B
C
-4
-6
-8
-10
A
D
-12
-14
-16
-18
A: (-10.02, -11.32) B: (-4.97, -2.68) C: (5.04, -2.50)
f(x) = -0.11⋅x2
D: (9.98, -11.04)
-20
-22
-24
-26
-28
2
x
f(x)
-10.02
-11.32
-8.01
-6.98
-4.97
-2.68
-3.99
-1.73
-3
-1.02
-2.01
-0.53
0
0
3
-0.63
4.02
-1.45
5.04
-2.5
7.97
-6.98
9.98
-11.04
I plotted several more points in Geometer’s SketchPad and had it
measure the coordinates for me.
I then entered these data in Excel Spreadsheet and fitted a quadratic
“trendline.”
I also had it calculate the R2 value, which is nearly 1, which means
that the fit is very good (R=0.9994).
f(x) y = -­‐0.1126x2 + 0.0127x + 0.1502 R² = 0.9988 2 0 -­‐15 -­‐10 -­‐5 0 5 10 15 -­‐2 f(x) -­‐4 Poly. (f(x)) -­‐6 -­‐8 -­‐10 -­‐12 3
Additional Information:
Filmy Dome Web: From: http://www.pestproducts.com/spider-webs.htm
And http://hartmanprehistoricgarden.com/sa-neriene.html
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The Filmy Dome Spider is a very common species that tends to build its web in shady,
protected areas. These spiders are only about ¼ inch long. A dorsal view of the female is
shown in the photo above.
This species obviously gets its common name from the web in which it lives. There is a
sort of dome, along with a jumble of cobwebs all around it. However, it is almost
invisible in most lighting situations. When the sun hits it at the right angle, and from a
very low perspective, the dome becomes more apparent and the web shines with rainbow
hues.
From: http://www.biokids.umich.edu/guides/tracks_and_sign/build/webs/
Sheet web spiders (Linyphiidae) make many kinds of webs that are formed out of sheets
of silk. The sheets are wildly jumbled together and don't have many large gaps.
There are several kinds of sheet web spiders. Platform spiders make thickly interwoven
sheets of silk. Filmy dome spiders make dome shaped sheets that are secured by a
network of silk strings. Bowl and doily spiders make very interesting webs. The tops of
them look like bowls that are secured by strings to something above them. The bowl
appears to be sitting on a doily that is the second part of the web and is attached
horizontally. These kinds of webs make very effective booby-traps for unsuspecting
insects.
From:
http://bugguide.net/node/view/323460/bgimage
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From:
http://www.sergeev.us/pictures/archives/compress/2009/721/24s.htm
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