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Investigate the mathematical functions which model these curves and determine Logans logo

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Introduction

In this assignment, I will investigate the mathematical functions which model these curves and determine Logan’s logo.

For this, I will use an appropriate set of axis with a scale of 1 centimetre per unit and record a number of different data points on the curves which will then create model functions.

Logan’s logo represents two curves: the lower curve and the upper curve. The variables used are the horizontal-axis and the vertical-axis.

The measured numbers of data points for the lower and upper curve are represented in this table.

1. The various numbers of data points from the curve are entered in a table as shown below. L1 being the horizontal-axis, L2 the lower curve and L3 the upper curve.

image07.png

The window parameters for the graph are then programmed for both the upper and the lower curve to respect the scale on Logan’s logo:

image08.png

Then a scatterplot is obtained for both curves after putting the points of each curve into a table.

image17.png

Graph of the lower curve:

image22.png

...read more.

Middle

Using the sinusoidal regression formula the values of a, b, c and d for the upper sine curve are found:

image11.png

Following the sine formula, the letters are replaced with the values found with the sinusoidal regression formula in the function plot to then create a graph for each curve.

image12.png

The graph of the lower sine curve resembles this:

image13.png

The same method is replicated for the second upper sine curve:

image14.png

The graph of the upper sine graph is represented as the following:

image15.png

The amplitude of the upper sine curve is calculated with the following formula: image16.png

However, the sine curves of Logan’s logo carry some limitations.

First of all, it is not certain that this curve follows the sinusoidal behaviour. It is only a possible hypothesis. Furthermore, the values are not extremely precise therefore the results of the sine equations are not very accurate. Thirdly, as it was graphically impossible to obtain the correlation coefficient (r),

...read more.

Conclusion

image09.png5.70

The amplitude is measured with the following formula:

Amplitude=image16.png=

image23.png

ower limit=0

Upper limit=8

3. For the business card, we must multiply the amplitude by 5/8 and to find the area between two curves, we find the area under the curve which is above and subtract the area of the curve which is below.

The lower limit is 0 and the upper one is 8.  (screenshot 2nd calc , 7)

In order to find the area between the two curves, we have to substract the area of the lower curve to the area of the upper curve.

image18.png32.13-20.15image24.png

The business card’s area is 45cm2. The amplitude of the two sine functions must be multiplied by image25.png to reduce the height (y-axis) to 5 cm

Logan’s logo occupies image26.png of the business card.

Pour la 1ère function A has to be multiplied by 5/8 and d which affects vertical translation  ce qui donne 1.27 et d= 2.42*5/8=1.51

Pour la 2ème function a (amplitude) and the d by 5/8 2.93 * 5/8=1.89. et d=4.03*5/8=2.52

Screenshots of business-card format, sine functions :

Then calculate the integrals and area

Sine function 1: aimage21.png

image27.png for the lower curve

For the upper curve f(x)=2axsin(bx+c)d

image28.png cm 2 for the upper curve.

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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