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Algrebra:Gatwick airport temperatures

Extracts from this document...

Introduction

807617.doc                                      

AS Use of Maths

Working with algebraic and graphical techniques

Assignment 2: Gatwick airport average temperatures

Introduction

Data:

image00.png

Graph:

image01.png

This is cyclical and will there for be modelled by a trigonometrical function

of the form

Y    =  A  sin (ω(t +  α)) + C

where

A is the amplitude

ω is the frequency

t is the time in days

α is the phase shift

C is the central line


Finding the model

The central line is:  (min + max)/2 = (3.8+16.5)/2 = 10.15

...read more.

Middle

The initial estimate for the phase shift was 110 to the right, image03.png

so by translation t → (t – 110). This result was not on the original data, but further modifications gave rise to t → ( t -113) which appeared closest.

On checking the values for the average absolute error for the points given by the data, this also gave the smallest error.

So y = 6.35snin (0.986(t – 113)) + 10.15 is the model for the data

Checking the model works

The data give (197, 16.5)

When t = 197, the model gives

y = 6.6 sin (0.986(197 – 113)) + 10.2

y = 6.6 sin (82.824)+10.2 = 6.6 x 0.992 + 10.2 = 16.75 is very close to the data value of 16.

...read more.

Conclusion

0C, when will that be:

For y = 6.35snin (0.986(t – 113)) + 10.15, when y = 10 then

10 = 6.35 sin (0.986(t – 113)) + 10.15

Subtracting 10.15 from both sides

                                -0.15 = 6.35snin (0.986(t – 113))

Dividing both sides by 6.35

                                -0.15/6.35 = -0.024 = sin (0.986(t – 113))

Taking the inverse sine of both sides

                                sin-1(-0.024) =  (0.986(t – 113))

-1.35 = 0.986(t – 113)

Divide by 0.986                -1.35/0.986 =  -1.3692 = t – 113

Add 113                        -1.3692 + 113 = 111. 63 = t

This means on the day which is the heating will be turned off is 112 day which is 22 April

Evaluation and conclusion

The model works well in mirroring the data. Unless climate change really increases, it is likely that this model will be useful for some years to come

LMcG                

...read more.

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