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# Logan's logo

Extracts from this document...

Introduction

Math Portfolio

Type 2

Logan’s Logo

Class: IB07

Introduction

Logan has designed the logo at the right. The diagram shows a square which is divided into three regions by two curves. The logo is the shaded region between the two curves. Logan wishes to find mathematical functions that model these curves. Now in order to find this mathematical function, the square needs to be measured by placing it on a graph paper (figure 2) and then the data points must be identified and recorded to represent a model function for each of the curves. I have decided to choose my units in cm.

Figure 1

From this simple plot we can estimate and examine a proper set of axes and record the coordinates in each curve.

Middle

5.7

5.5

6.3

6

6.55

6.3

6.6

6.5

6.5

7

6.1

7.5

5.1

Variables:

• x represents any data point on the x-axis in the domain (0-7.5) which stand for the lower curve
• f(x) is the image for all the values of x and is permitted in a range of  0.7- 4.23
• u represents any data point on the x-axis in the domain (0-7.5) which stand for the upper curve
• f(u) is the image for all the values of x and is permitted in a range of 2.5- 6.6

a graph has been drown to display the data points shown in table 1 and 2

The data points in the tables have been applied to model functions that follow the behavior of each curve shown. We can notice from the graph that these two curves are parallel to each other hence they might follow the same type of equation but vary in the parameters values.

Technology (GDC calculator) has been used to plot these curves and many types of functions have been examined to this model. I have come to the conclusion that the functions type that models the data behavior is a quartic equation.

Quartic equation = ax4+bx3+cx2+dx+e

Conclusion

f(x)= ax4+bx3+cx2+dx+e:

w.f(zx)=w[a(zx)4+b(zx)3+c(zx)2+dzx+e]  = wa(zx)4+wb(zx)3+wc(zx)2+wdzx+ we

Business card with the logo extended on it

The fraction area can be measured by measuring the area under each of the curves and subtracting the area under the upper curve fron the area under the lower curve. This is how the area of the logo can be calculated

To measure the area under each of the curves the definite integral is taken for each of the graphs of the functions f(x) and f(u)  where there is a lower limit and upper limit.

The equation of the lower curve in the card is:

f(x)= 0.0013178775(0.83333x)4 – 0.0637028755(0.83333x)3 +0.5254143882(0.83333x)2 – 0.724055755(0.83333x) + 0.5214647399=0

The equation of the upper curve in the card is:

f(u) = – 4.00148714x10-4(0.83333u)4 – 0.656078266(0.83333u)3 + 0.7829361043(0.83333u)2 – 1.765631742(0.83333u) + 1.793820961 = 0

the general equation after integration:

*Where the constant b of the integral in the lower limit

*the constant a of the integral is the upper limit

The area under the lower graph is:

= 10.74054666cm2

The area under the upper graph is:

= - 174.21259 < there is probably something wrong with the calculating

The accuracy of area of the logo is important in a business card because there are always some code and numbers written within these logos.

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

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