SSS 2 SECOND TERM MATHEMATICS WEEK SEVEN

**Topic: ****BEARING AND DISTANCES **

Bearing can be defined as the clockwise angular movement between two distant places.

**Rules for Solving Bearing and Distances**

1. Taking reading in bearing starts from the North Pole in clockwise direction and ends also at the North Pole. I.e. North – North Pole reading.

2. All angles formed while taking reading in bearing is equal to 360 degrees.

3. All questions in bearing leads into the formation of a triangle.

**Examples;**

1. A boat sails 6km from a port X on a bearing of 065^{0} and thereafter 13km on a bearing of 136^{0}. What is the distance and bearing of the boat from X?

**Solution**

**Step 1**

You will need to represent the question on a triangle.

You will actually need a ruler and pencil for drawing along with a good eraser in case you make mistake.

Take your pencil and draw a point X, take ruler and rule a straight line to another point Y and rule another straight line to point Z and finally close it up at point X again. See below;

**Step 2**

Use Cosine Rule to solve for y km;

**From Cosine Rule;**

b^{2} = a^{2} + c^{2} – 2ac Cos B

**Step 3**

Replace a, b, c and B with x, y, z and Y respectively;

y^{2} = x^{2} + z^{2} – 2xz Cos Y

y^{2} = 13^{2} + 6^{2 }– 2 (13) (6) Cos 109^{0}

y^{2} = 169 + 36 – 156 (- 0.3256)

**Note; (- × – = +)**

y^{2} = 205 + 156 x 0.3256

y^{2} = 205 + 50.7886

y^{2} = 255.7886

**Step4**

Take square root of both sides;

y = √255.7886

y = 15.9934km app. 16km (to the nearest km).

**Step5**

To find the bearing of the boat from X, we will apply Sine Rule;

**From Sine Rule;**

Sin X/x = Sin Y/y

Sin θ/13km = Sin 109^{0}/16km

**Step 6**

Cross Multiply;

16km x Sin θ = 13km x Sin109^{0}

16km x Sin θ = 13km x 0.9455

16km (Sin θ) = 12.2917

**Step 7**

Divide both sides by 16km;

Sin θ = 12.2917/16km

Sin θ = 0.7682

θ = Sin^{-1 }0.7682

θ = 50.1955^{0 }app. 50^{0 }(to the nearest degree)

**Step 8**

The bearing of boat from the port will be 065^{0} + θ

Which is 065^{0} + 50^{0} = 115^{0} (answer).

2. An aircraft takes off from an airstrip at an average speed of 20km/hr on a bearing of 052^{0} for 3 hours. It then changes course and flies on a bearing of 028^{0} at an average speed of 3okm/hr for another 1^{1}/_{2} hours. Find; a. its distance from the starting point, b. the bearing of the aircraft from the airstrip.

**Solution**

**Step 1**

**Step 2**

Use Cosine rule to solve for side b;

**From Cosine Rule;**

b^{2} = a^{2} + c^{2} – 2ac Cos B

b^{2} = 45^{2} + 60^{2} – 2 (45) (60) Cos 156^{0}

b^{2 }= 2025 + 3600 – 5400 (- 0.9135)

b^{2} = 5625 + 5400 x 0.9135

b^{2} = 5625 + 4932.9

b^{2} = 10557.9

b = √10557.9

b = 102.7516 app. 103 (nearest whole number)

b = 103km

**Step 3**

To find the bearing of the aircraft from the airstrip, you must first find θ;

Find θ using Sine Rule below;

**From Sine Rule;**

Sin A/a = Sin B/b

Sin θ/45km = Sin 156^{0/}103km

**Step 4**

Cross Multiply;

103km (Sin θ) = 45km x Sin 156^{0}

**Step 5**

Divide both sides by 103km;

103km (sin θ)/103km = 45 x 0.4067/103km

**Note;** (103km cancels itself and 45 x 0.4067 gives 18.3015)

Sin θ = 18.3015/103

Sin θ = 0.1776

θ = Sin^{-1} 0.1776 (**Note;** when taking Sin to the RHS, it becomes inverse)

θ = 10.2299^{0}

To find the bearing will be 052^{0} – θ

Bearing = 052^{0} – 10.2299^{0}

Bearing = 41.77^{0}

**Practice these questions below;**

1. A ship leaves port and travels 21km on a bearing of 032^{0} and then 45km on a bearing of 287^{0}.

(a) Calculate its distance from the port.

(b) Calculate the bearing of the port from the ship.

2. A surveyor leaves her base camp and drives 42km on a bearing of 032^{0}, she then drives 28km on a bearing of 154^{0}. How far is she then from her base camp and what is her bearing from it?

3. Two ships leave port at the same time and travels at 5km/hr on a bearing of 046^{0}. The other travels at 9km/hr on a bearing of 127^{0}. How far apart are the ships after 2 hours?

4. A triangular field has two sides 50m and 60m long, and the angle between them is 096^{0}. How long is the third side?