
UNIT 6 : BASIC
TRIGONOMETRY WITH TRIANGLES
LESSON
5: PROBLEM SOLVING HOMEWORK
QUESTIONS (Solutions below)
1. To find the width across a river, Tina marks
off a baseline distance PQ of 150
m. The angles of sight to a point R on
the opposite side of the river are
< P = 61^{0} and < Q = 44^{0}
as shown in her sketch below. Calculate
the width of the river (RM).
2. From the top of a fire tower 190 m high, the
angle of depression of a fire in the distance is 22.6^{0}. Find the distance of the fire from the
bottom of the tower.
3. From the top of a 123 m cliff, the angles of
depression of ships Q and R at sea are 21^{0} and 35^{0}
respectively. Find the distance (QR)
between them.
4. Somaiah
travels due north for 2.5 h at 90 km/h.
From there she travels N60^{0}E at 100 km/h for 3h. How far is she from her starting point.
5. Find the measure of the marked angle (< A
) to the nearest tenth of a degree.
A B
39.7^{0}
8.2 m
C
7.5 m 7.8 m
41.8^{0}
D
E
6. Two roads leave from Cairo at an angle of 38^{0}
to each other. One road goes to Alford
and the other to Bargaintown. The road
from Alford to Bargaintown [AB] is 22.6 km and meets the road from Cairo to
Bargaintown [CB] at an angle of 53^{0}. Find the distances from Cairo to Alford [CA]
and From Cairo to Bargaintown [CB].
A
22.6 km
38^{0} 53^{0}
C B
7. A surveyor wishes to find the height of a
mountain PS in the diagram. The angle of elevation of the top of the mountain
from point Q is 14.2^{0}. A
base line QR of length 452 m is staked out and angles SQR and SRQ are shown in
the diagram. Find the height h
of the mountain.
8. Amane wishes to find the width of a marsh
AB. She takes measurements as shown in
the diagram. Find the width AB.
9. Two cars leave Parkersburg at the same
time. Car W travels to Wheeling, a
distance of 85 km. Car S travels to
Strasburg, a distance of 238 km. The
angle between the roads they take is 56.4^{0}. Find the distance between Wheeling and
Strasburg.
10. Two blimps are located directly over a
straight road. The angles of elevation
of blimp A from points P and Q are 70^{0} and 22^{0}
respectively. The angles of elevation
of blimp B are 65^{0} and 24^{0} as shown in the diagram. How far apart are the balloons to the
nearest tenth of a km? The distance PQ
is 4.7 km.
11. Frieda travels due west at 95 km/h for 3
hours. Ahmed travels N30^{0}W
at 100 km/h for 2 ˝ hours. How far
apart are they if they started from the same location?
12. From the top of a lighthouse [A] 40 m high,
the angles of depression of two ships [B and C] at sea are 34^{0} and
32^{0}. The angle between the
lines of sight with the bottom of the lighthouse is 105^{0}. Find the distance between the two ships
[BC].
Solutions:
1. To find the width across a river, Tina marks
off a baseline distance PQ of 150
m. The angles of sight to a point R on
the opposite side of the river are
< P = 61^{0} and < Q = 44^{0}
as shown in her sketch below. Calculate
the width of the river (RM).
Solution:
2. From the top of a fire tower 190 m high, the
angle of depression of a fire in the distance is 22.6^{0}. Find the distance of the fire from the
bottom of the tower.
Solution: The angle between the horizontal (dashed line AD) and
the line of sight (AC) of the fire is the ANGLE OF DEPRESSION ( <DAC in
diagram)
A
D
)Angle of depression =
22.6^{0}
190 m
B x C
3. From the top of a 123 m cliff, the angles of
depression of ships Q and R at sea are 21^{0} and 35^{0}
respectively. Find the distance (QR)
between them.
Solution:
4. Somaiah
travels due north for 2.5 h at 90 km/h.
From there she travels N60^{0}E at 100 km/h for 3h. How far is she from her starting point.
Solution:
90 km/h
x 2.5 h = 225 km due north.[BA]
100 km/h
x 3 h = 300 km N60^{0}E [AC]
N
5. Find the measure of the marked angle (< A
) to the nearest tenth of a degree.
A B
39.7^{0}
8.2 m
C
7.5 m 7.8 m
41.8^{0}
D
E
Solution:
6. Two roads leave from Cairo at an angle of 38^{0}
to each other. One road goes to Alford
and the other to Bargaintown. The road
from Alford to Bargaintown [AB] is 22.6 km and meets the road from Cairo to
Bargaintown [CB] at an angle of 53^{0}. Find the distances from Cairo to Alford [CA]
and From Cairo to Bargaintown [CB].
A
22.6 km
38^{0} 53^{0}
C B
Solution:
This is the AAS case. Use the sine law to find CA first.
7. A surveyor wishes to find the height of a
mountain PS in the diagram. The angle of elevation of the top of the mountain
from point Q is 14.2^{0}. A base
line QR of length 452 m is staked out and angles SQR and SRQ are shown in the
diagram. Find the height h of
the mountain.
Solution:
Again
use the triangle with 3 elements known.
This would be triangle QSR.
This is the AAS case. Use the sine law.
8. Amane wishes to find the width of a marsh
AB. She takes measurements as shown in
the diagram. Find the width AB.
Solution:
This is the SSA case. Use the sine law.
9. Two cars leave Parkersburg at the same
time. Car W travels to Wheeling, a
distance of 85 km. Car S travels to
Strasburg, a distance of 238 km. The
angle between the roads they take is 56.4^{0}. Find the distance between Wheeling and
Strasburg [WS].
Solution:
W
85
56.4^{0}
P 238 S
This is
the SAS case. Use the
cosine law for side p.
Therefore
the distance between Wheeling and Strasburg is 203.7 km.
10. Two blimps are located directly over a
straight road. The angles of elevation
of blimp A from points P and Q are 70^{0} and 22^{0}
respectively. The angles of elevation
of blimp B are 65^{0} and 24^{0} as shown in the diagram. How far apart are the balloons to the
nearest tenth of a km? The distance PQ
is 4.7 km.
Solution:
11. Frieda travels due west at 95 km/h for 3
hours. Ahmed travels N30^{0}W at
100 km/h for 2 ˝ hours. How far apart
are they if they started from the same location?
Solution:
Frieda travels a distance of 95 x 3 = 285 km.
due west. Ahmed travels 110 x 2.5 = 275
km at an angle of 60^{0} to Frieda’s path.
This is the SAS case. Use the cosine law for side AB.
12.
From
the top of a lighthouse [A] 40 m high, the angles of depression of two ships [B
and C] at sea are 34^{0} and 32^{0}. The angle between the lines of sight with the bottom of the
lighthouse is 105^{0}. Find the
distance between the two ships [BC].