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# FM Sphere

## 2.9 Introduction to Time Zones

Note: in Further Maths, it is expected that you express time in the form hrs:mins hours e.g. 1:30 hours.

### Time Zones

• As the sun orbits around the world, different parts of the world will experience a different “part of the day”. While some are in broad daylight, others are in complete darkness. We want the time of day to tell us, regardless of where we are in the world, how long we have left before the sun rises or sets. To allow this, time zones are created. Times zones are regions which share a common time. Different time zones are set to different times to give a depiction of when the sun is expected to rise/set in that location.
• Time zones separated longitude. In Further Maths, we will approximate that each time zone is separated by 15o of longitude and each subsequent time zone differs by 1 hour.
• Time zones are set relative to Greenwich Mean Time (GMT), the time zone corresponding to the Greenwich meridian (0o). Locations east of the meridian are ahead of GMT, while locations west of the meridian are behind of GMT.

### Time in a different Time Zone

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## 2.8 Shortest Distance between Points on a Parallel

Note: if you cannot remember what parallels of longitude are, revise notes for 2.6 Modelling the World.

Note: if you cannot remember how to find the shortest distance between points on great and small circles, revise notes for 2.5 Distance Between Points on a Sphere.

### Shortest Distance between Points on the Equator

• Locations of equal latitude exist on the same parallel of longitude. By taking the shorter section of that parallel with the locations as the endpoints, we can extract an arc representing the shortest path between the two locations. The length of that path can be found using the longitudinal coordinates of the locations.
• The plane of the equator intersects the centre of the world and so it is a great circle.
• We can modify the equation for arc length (from notes 2.1 Circles and Arcs) for this situation. There are 3 situations with different formulas:
• If both locations are given in the same units (oE or oW) or one or both lie on the Greenwich meridian, the formula is:
Read More »2.8 Shortest Distance between Points on a Parallel

## 2.7 Shortest Distance to the Poles or Equator

Note: if you cannot remember what a meridian is, revise notes for 2.6 Modelling the World.

### Finding the Shortest Distance using the Meridian

• The shortest path between any point on the surface of the earth to the north or south pole, or the equator is along the meridian intersecting that point. By taking the section of the meridian between the relevant pole or equator and the point being solved for, we find an arc whose length is the shortest distance between the pole or equator and the position. The angle of that arc can be found using the latitude of that location.
• We can modify our formula for arc length (from 2.1 Circles and Arcs) to be specific for this situation:

s_{\text {north pole }}=\frac{\pi 6400\left(90-l_{N}\right)}{180}=\frac{\pi 640\left(90-l_{N}\right)}{18}

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## 2.6 Modelling the World

### The World as a Sphere

• The world can be approximated to be a sphere of radius 6400km.

### Meridians

• Meridians of longitude are circles drawn around the world with the same radius as the world and intercepting the north and south poles.

Example

In the above spherical representation of the world, a meridian of longitude is shown in red.

### Parallels

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## 1.2 Volume of Common Shapes

Note: volume is measured in units cubed (e.g. m^3, cm^3, mm^3, etc.).

### Volume of a Sphere

• The volume of a sphere is found using the formula:

V=\frac{4}{3} \pi r^{3}

Example

We wish to find the volume of the above sphere. We do this by first identifying the values we need, in this case we have a radius of 10mm. Now, we substitute this value into our formula to find the volume

V=\frac{4}{3} \pi r^{3}=\frac{4}{3} \pi 10^{3}=4188.79 \mathrm{~mm}^{3}

### Volume of a Cylinder

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## 1.1 Surface Area of Common Shapes

Note: remember that area is measured in units squared (m^2, cm^2, etc.).

### Surface Area of Spheres

• The surface area of a sphere is found using:

A=4 \pi r^{2}

Where r is the radius of the sphere.

Example

We wish to find the surface area of the above sphere. We do this by first identifying the values we need, in this case we have a radius of 10mm. Now, we substitute this value into our formula to find the surface area:

A=4 \pi r^{2}=4 \pi 10^{2}=1256.64 \mathrm{~mm}^{2}

### Surface Area of Cylinders

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