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• Level: GCSE
• Subject: Maths
• Word count: 1010

# A group of researchers set up a series of observation sites in the Arctic on a circumference of a circle of radius 50km.They need to set up a base camp so they can visit a different site each day.

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Introduction

## Arctic Research

Aim

A group of researchers set up a series of observation sites in the Arctic on a circumference of a circle of radius 50km.They need to set up a base camp so they can visit a different site each day.

Assumptions

These are assumptions that can be made.

1. Altitude
2. Weather conditions
3. Air resistance
4. Plane speed constant
5. Wind speed constant throughout/No gust/Same direction
6. Time for turning around
7. Time for taking off and landing
8. Rotation of the Earth
9. Weight of fuel
10. Thermal
11. Safety
12. Breakdowns (damaged plane)
13. Plane could get damaged and not fly as well as before, result in less speed and longer journey time.

Model 1

To start with a simple model would be for a plane to fly from A to B with no wind speed and ignoring all the assumptions as stated above.

A 50 km B
---------------------------------------------------------------------

Plane = 100 mph Wind speed = 0
= 100*1.6
= 160 kmh-1

Time = Distance/Speed
= 50/160
= 18.75 minutes

For the whole journey A to B then back to A would be:
Time from A to B multiplied by 2. 2*18.75 = 37.5 minutes
Equation
Time (A to B to A) = 2(Distance/Speed)
OR

Middle

The rest of the assumptions stated above are ignored in this model for simplicity.

Evaluation

When the resultant speed was calculated for right angle triangles Pythagoras´ theorem was used, bearings were found by using Trigonometry (Sin = opposite/hypotenuse Cosine = adjacent/hypotenuse Tangent = opposite/adjacent). For triangles that were not right angled they were drawn to scale.
A way of finding resultant speeds quicker would be to only calculate angles from one quarter of the circle and then use symmetry in the y and x-axis. With angles such as 30 degrees a mirror line could be used in the y = -x axis to find the resultants flying back in the opposite direction. Symmetry in the y-axis would get you the opposites of that angle in which the two triangles are drawn together side by side as shown in my scale drawings for 45-degrees.

The reason for not having time for landing and taking off is because the times would all be the same just with 10-20 minutes or however long it takes for the plane to take off and land added onto the resultant time.

Conclusion

More work I could have done was to investigate to see how long journeys would take if the plane had time for taking off and landing or if the plane was to just circle around at the destination and go back to base. The assumption is that it would be easier to model the plane landing and taking off rather than flying around and then back to base because if it was to fly around you´d have to model if it was to do a circular or an ellipse turn. The circular turn would be easier to model than the ellipse.

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To start with a simple model would be for the plane to fly from A to B with no wind speed and ignoring all the assumptions as stated above. A 50 km B --------------------------------------------------------------------- Plane = 100 mph Wind speed = 0 = 100 x 1.6 = 160 kmh-1 Time

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