Home Right Triangles Angles in Standard Position Sine Law & Ambiguous Case Cosine Law Problem Solving Summary&Test

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 = 610 and < Q = 440 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 22.60.  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 210 and 350 respectively.  Find the distance (QR) between them.

4.  Somaiah  travels due north for 2.5 h at 90 km/h.  From there she travels N600E 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

39.70

8.2 m

C

7.5 m                          7.8 m

41.80

D

E

6.  Two roads leave from Cairo at an angle of 380 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 530.  Find the distances from Cairo to Alford [CA] and From Cairo to Bargaintown [CB].

A

22.6 km

380                                       530

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 14.20.  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 56.40.  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 700 and 220 respectively.  The angles of elevation of blimp B are 650 and 240 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 N300W 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 340 and 320.  The angle between the lines of sight with the bottom of the lighthouse is 1050.  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 = 610 and < Q = 440 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 22.60.  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 = 22.60

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 210 and 350 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 N600E 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 N600E [AC]

N

5.  Find the measure of the marked angle (< A ) to the nearest tenth of a degree.

A                                                           B

39.70

8.2 m

C

7.5 m                          7.8 m

41.80

D

E

Solution:

6.  Two roads leave from Cairo at an angle of 380 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 530.  Find the distances from Cairo to Alford [CA] and From Cairo to Bargaintown [CB].

A

22.6 km

380                                       530

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 14.20.  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 56.40.  Find the distance between Wheeling and Strasburg [WS].

Solution:

W

85

56.40

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 700 and 220 respectively.  The angles of elevation of blimp B are 650 and 240 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 N300W 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 600 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 340 and 320.  The angle between the lines of sight with the bottom of the lighthouse is 1050.  Find the distance between the two ships [BC].