Real-Life Math
Entomologists study insects and everything about them, including
their habitats, their social structure, how they reproduce and what they eat.
Because there are often many different species of one type of insect, it's
not uncommon for an entomologist to specialize in one particular insect group.
You're
an entomologist. You have been working several months with dragonfly larvae,
trying to come up with a method to accurately determine the different species
of larvae. As many entomologists know, adult dragonflies are relatively easy
to differentiate. Dragonfly larvae, however, are a different story.
"Often
you know that the dragonfly larvae you're researching are different species,"
says entomologist Karen Needham. "But you can't see anything obvious [to tell
them apart]."
So researchers must try to find a pattern and create
a key, adds Needham. That often means developing a ratio that can be used
to identify 1 species from another.
Recently, you've been spending
your time distinguishing between 2 different types of dragonfly larvae. By
comparing the width of an insect's head to the length of its abdomen, and
by measuring the width of its eye and comparing it to the space between the
eyes, you think you've finally determined the ratios to identify and separate
these 2 particular species.
Out of the 100 insects you've tried these
measurements and ratios on so far, about 40 percent of the insects are categorized
as species A. About 60 percent have been categorized as species B.
You
decide to measure 3 more insects and compare ratios, just to make sure your
theory can be considered "foolproof."
These are the ratios you have
come up with to determine the 2 different species:
Species A
insects: the head width ratio in comparison to abdomen length is a 2:5 ratio
(or 0.4). So if an insect's head were 2 inches wide, the length of its abdomen
would be 5 inches.
These same insects must have an eye width to "space
between the eyes" ratio of 1:4 (or 0.25). This means if an insect's eye width
was 0.1 inches, the length between the eyes would be 0.4 inches.
Insects
categorized as species B would have slightly different
ratios. Insects would need to have a head width to abdomen length ratio of
4:5 (or 0.8).
They would also need an eye width to space between the
eyes ratio of 2:3 (or 0.66).
The 3 insects you need to categorize have
the following measurements:
Insect 1 -- has a head that measures
2 inches in width and an abdomen that measures 2.5 inches in length. It also
has an eye that measures 0.2 inches in width and the space between its eyes
measures 0.3 inches.
Insect 2 -- has a head that measures 4
inches in width and an abdomen that measures 10 inches in length. Its eye
width measures 0.3 inches and the space between its eyes measures 1.2 inches.
Insect
3 -- has a head that measures 1.5 inches in width and an abdomen that
measures 2.5 inches in length. Its eye width is 0.6 inches and the space between
its eyes is 0.9 inches.
You need to determine which insect(s) should
be categorized as species A and which insect(s) should be categorized as species
B.