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Chapter Projects
C
Diagram i
Q
B
A
(c) How far did the explorers travel just to get that far?
(d) Draw a plane triangle connecting the three towns. If
the distance from Lewiston to Great Falls is 282 miles
and the angle at Great Falls is 42° and the angle at
Lewiston is 48.5°, find the distance from Great Falls
to Lemhi and from Lemhi to Lewiston. How do these
distances compare with the ones computed in parts
(a) and (b)?
American Journey: The Quest for Liberty to 1877,
Texas Edition. Prentice Hall, 1992, p. 345.
National Geographic Atlas of the World, published by
National Geographic Society, 1981, pp. 74–75.
a
b
O
c
P
(e) Replacing the ratios in part (d) by the cosines of the
sides of the spherical triangle, you should now have
the Law of Cosines for spherical triangles:
cos C = cos A cos B + sin A sin B cos C
Mathematics from the Birth of Numbers by Jan Gullberg. W.W. Norton & Co., Publishers, 1996, pp. 491–494.
B. Lewis and Clark followed several rivers in their trek
from what is now Great Falls, Montana, to the Pacific
coast. First, they went down the Missouri and Jefferson
rivers from Great Falls to Lemhi, Idaho. Because the two
cities are on different longitudes and different latitudes,
we must account for the curvature of Earth when computing the distance that they traveled. Assume that the
radius of Earth is 3960 miles.
(a) Great Falls is at approximately 47.5°N and 111.3°W.
Lemhi is at approximately 45.0°N and 113.5°W. (We
will assume that the rivers flow straight from Great
Falls to Lemhi on the surface of Earth.) This line is
called a geodesic line. Apply the Law of Cosines for a
spherical triangle to find the angle between Great
Falls and Lemhi. (The central angles are found by
using the differences in the latitudes and longitudes of
the towns. See the diagram.) Then find the length of
the arc joining the two towns. (Recall that s = ru.)
Diagram ii
North
b
Great
Falls
a
Lemhi
c
South
(b) From Lemhi, they went up the Bitteroot River and
the Snake River to what is now Lewiston and
Clarkston on the border of Idaho and Washington.
Although this is not really a side to a triangle, we will
make a side that goes from Lemhi to Lewiston/
Clarkston. If Lewiston and Clarkston is at about
46.5°N 117.0°W, find the distance from Lemhi using
the Law of Cosines for a spherical triangle and the
arc length.
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2.
Leaning Tower of Pisa PISA, Italy—Workers
began removing two sets of steel suspenders attached to
the leaning Tower of Pisa on Tuesday, in one of the final
phases of a bold plan to partially straighten the famously
tilted monument.
The 340-foot-long cables had been recurred to the
tower in 1998 as a precaution in case it needed to be
yanked back up while the soil under its foundation was
being excavated. Anchored to giant winches dug into the
ground about 100 yards from the tower, the suspenders
haven’t been needed.
Removal of the suspenders will be completed in time
for an inauguration ceremony of the newly straightened
tower scheduled for June 16, said Paolo Heiniger, who
oversees the project.
The tower was closed to the public more than a
decade ago, when officials feared it was beginning to lean
so much that it might topple over.Work to stop the tower’s
increasing tilt has taken far longer than planned, but
officials expect it to be open to tourists in the fall.
When work began, the tower leaned 6 degrees, or 13
feet, off the perpendicular on its south side. By removing
a small amount of soil, the tower has settled better and
now leans about 16 inches less—nearly the tilt it had 300
years ago.
The decrease in lean is not enough for the naked eye
to detect but sufficient to stabilize the monument, experts
have said.
Use the fact that the tower was 184.5 feet tall when it
stood upright to answer the following questions:
(a) The article states that when work began the tower
leaned 6°, or 13 feet, off the perpendicular. Draw a
sketch and label the two measurements.
(b) How high is the tower with a lean of 6°?
(c) The article goes on to say that the tower now leans 16
inches less. Draw a new sketch that shows this measurement.
(d) What is the degree measure of this lean?
(e) What is the height of the tower with this lean?
13
(f) Comment on the fact that sin 4° =
= 0.07046,
184.5
while sin 6° = 0.10453. Can you reconcile this seeming discrepancy?
(g) Investigate further and write a report on your findings.
Source: Naples Daily News,Wednesday, May 16, 2001.
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3.
CHAPTER 9
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Applications of Trigonometric Functions
Locating Lost Treasure While scuba diving off
Wreck Hill in Bermuda, a group of five entrepreneurs discovered a treasure map in a small watertight cask on a pirate schooner that had sunk in 1747. The map directed
them to an area of Bermuda now known as The Flatts.
The directions on the map read as follows:
1. From the tallest palm tree, sight the highest hill. Drop
your eyes vertically until you sight the base of the hill.
2. Turn 40° clockwise from that line and walk 70 paces
to the big red rock.
3. From the red rock walk 50 paces back to the sight line
between the palm tree and the hill. Dig there.
Highest point
Base of hill
50 paces
40°
Upon reaching The Flatts, the five entrepreneurs
believed that they had found the red rock and the highest
hill in the vicinity, but the “tallest palm tree” had long
since fallen and disintegrated. It occurred to them that
the treasure must be located on a circle with radius 50
“paces” centered around the red rock, but they decided
against digging a trench 942 feet in circumference, especially since they had no assurance that the treasure was
still there. (They had decided that a “pace” must be about
a yard.)
(a) Determine a plan to locate the position of the lost
palm tree.
(b) One solution follows: From the location of the palm
tree, turn 40° counterclockwise from the rock to the
hill, then go about 50 yards to the circle traced about
the rock. Verify this solution.
(c) This location did not yield any treasure. Find the other
solution and the treasure.Where is the treasure? How
far is it from the palm tree?
Red
rock
70 paces
Palm
tree
Cumulative Review
1. Find the real solutions, if any, of the equation
3x2 + 1 = 4x.
9. Solve the triangle:
2. Find an equation for the circle with center at the point
1-5, 12 and radius 3. Graph this circle.
15
b
3. What is the domain of the function
f1x2 = 2x2 - 3x - 4?
20
4. Graph the function y = 3 sin1px2.
5. Graph the function y = -2 cos12x - p2.
3p
6. If tan u = -2 and
6 u 6 2p, find the exact value of:
2
(a) sin u
(b) cos u
(c) sin12u2
1
1
(d) cos12u2
(e) sina u b
(f) cosa ub
2
2
7. Use a graphing utility to graph each of the following
functions on the interval [0, 4]:
(a) y = ex
(b) y = sin x
(c) y = ex sin x (d) y = 2x + sin x
(b) y = x2
(e) y = ex
(h) y = cos x
10. In the complex number system, solve the equation
3x5 - 10x4 + 21x3 - 42x2 + 36x - 8 = 0
11. Analyze the graph of the rational function
R1x2 =
2x2 - 7x - 4
x2 + 2x - 15
12. Solve 3x = 12. Round your answer to two decimal places.
13. Solve log31x + 82 + log3 x = 2.
14. Suppose that f1x2 = 4x + 5 and g1x2 = x2 + 5x - 24.
8. Sketch the graph of each of the following functions:
(a) y = x
(d) y = x3
(g) y = sin x
40°
(c) y = 1x
(f) y = ln x
(i) y = tan x
(a)
(c)
(e)
(g)
Solve f1x2
Solve f1x2
Solve g1x2
Graph y =
= 0.
= g1x2.
… 0.
g1x2.
(b) Solve f1x2 = 13.
(d) Solve f1x2 7 0.
(f) Graph y = f1x2.