If you are more than 12 times zones west of Greenwich, the convention is to
start counting (at Greenwich) to the east, so we talk about 120°E longitude,
not 240°W. However, to follow the instructions here, figure it out going east
and add 360°. For example, if you followed these steps and got 242°W, that’s
the same as 118°E.
SUMMARY AND LEARNING MORE
In this chapter, we applied concepts from previous chapters to think about
the area of a circle. We spent some time thinking about the biggest polygons
that could fit into a circle, and the smallest ones that could fit around a circle,
and used those to estimate pi and understand where the formula for the area
of a circle came from.
This discussion of circles, plus our trigonometry basics in Chapter 6, gave
us the background we needed to be able to figure out our latitude and lon-
gitude based on a measurement one can do with a toothpick, a ruler, and a
sunny lunchtime. We will build on our latitude and longitude measurements
in Chapter 12 with more astronomy explorations. If you want to learn more
about why measuring longitude was so challenging, and the race to build
clocks to fix the problem, check out Dava Sobel’s book, Longitude (1995,
Walker Publishing).
Meanwhile, next up will be going into the third dimension. Chapter 8 will
explore the volumes of various solids, and how to use the ideas of volume
and density to see how Archimedes kept his king from being cheated.
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ANSWERS
Here are the answers for the activities in this chapter that we don’t solve in
the text of the section.
ARCTIC AND TROPICS
What is the maximum sun altitude at the South Pole(latitude -90°) on the first
day of southern summer (first day of winter in the north, with solar declina-
tion = -23.44°)? Where does that mean the sun is in the sky?
Answer: Maximum sun altitude = 90° - (-90°) - 23.44° = 180° - 23.44° =
156.56°. That number is hard to think about, so instead think of 180° as going
all the way from pointing due south, to pointing due north. We are going
23.44° less than that, so the sun would be 23.44° above due north at its
highest point on the southern summer solstice. To see what would happen
the rest of the day, check out the “Midnight Sun” article in Wikipedia.
Latitude 23.44° north is sometimes called the Tropic of Cancer, and the
equivalent in the south (at 23.44° south) is called the Tropic of Capricorn. On
the summer solstice in the northern hemisphere, what is the solar elevation
at local noon for a place on the Tropic of Cancer?
Answer: The sun would be directly overhead (at an elevation of 90°). The
definition of these latitudes is that it is the farthest north or south you can
go from the equator and have the sun be directly overhead on the first day of
local summer.
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