**FORM THREE MATHEMTICS STUDY NOTESTOPIC 7: THE EARTH AS THE SPHERE**

**THE EARTH AS THE SPHERE**

**Features and Location of Places**

Distance along Small Circles

Calculate distance along small circles

Suppose P and Q are places west or east of each other, i.e they lie on the same circle of latitude. Then when you travel due east or west from P to Q you travel along an arc of the circle of latitude.

The situation here is slightly different from that of the previous section. While circles of longitude all have the same length, circles of latitude get smaller as they get nearer the poles.

Consider the circle of latitude 50°S. Let its radius be r km.

**Nautical miles**

Example 8

Find the distance in km and nm along a circle of latitude between (20°N, 30°E) and (20°N, 40°W).

*Solution:*Both places are on latitude 20°N. The difference in longitude is 70°. Use the formula for distance.

Distance = 111.7 cos20° x 70°. Hence the distance in nautical miles is 60 x 70 x cos20°

The distance is

*3,950 nm.*Example 9

A ship starts at (40°S, 30°W) and sails due west for 1,000 km. Find its new latitude and longitude.

Example 10

A ship sails west from (20°S, 15°E) to (20°S, 23°E), taking 37 hours. Find speed, in knots and in kms per hr.

Exercise 4

Consider the following Questions.

*Navigation**Suppose a ship is sailing in a sea current, or that a plane is flying in a wind. Then the course set the ship or plane is not the direction that it will move in. the actual direction and speed can be found either by scale or by the use of Pythagoras’s theorem and trigonometry.*

*Draw the line representing the motion of the ship relative to the water. At the end of this line draw a line representing the current. Draw the third side of the triangle. This side, shown with a double – headed arrow, is the actual course of ship.*

Example 11

A ship sets course due east. In still water the ship can sail at 15km/hr. There is a current following due south of 4kkm/hr. use a scale drawing to find.

- The speed of the ship
- The bearing of the sip.

*Solution:*In one hour the ship sails 15km east relative to the water. Draw a horizontal line of length 15cm. In one hour the current pulls the ship 4km south. At the end of the horizontal line, draw a vertical line of length 4cm.

Example 12

The ship of example 10 needs to travel due east. Calculate the following.

- What course should be set?
- How long will the ship take to cover 120km?

*Solution*The ship needs to set a course slightly north of east, consider the following diagram.

**With no current, the journey would take 8hrs. The journey takes slightly longer when there is a current.Suppose a ship or a plane does not directly reach a position. We can still find how close the ship or plane is to the position.**

*Note:*Example 13

A small island is 200km away on a bearing of 075°. A ship sails on a bearing of 070°.Find the closest that the ship is to the island.

Exercise 5

1. Find the difference in longitude between Cape Town (34°S, 18°E) and Buenos Aires (34°S, 58°W)

2. A ship startsat (15°N, 30°W) and sails south for 2,500km. Where does it end up?

3. Find thedistance in km along circle of latitude between cape Town and Buenos aires (see question 2)

4. A plane starts at (37°S, 23°W) and flies east for 1,500 km. where does it end up?

5. Find thedistance in nau