Category: Mathematics and Statistics : Mathematics

You will be writing an International Baccalaureate Mathematics: Applications and Interpretations Higher Level Internal Assessment on using calculus to model the surface area of the whole building, the surface area of the Arabic writing around the building and volume of the Museum of the Future located in Dubai.
As this is a Higher Level IA most if not all of the math included in the paper will have to be HL math from the IB syllabus which I will provide. If needed you can still use standard level math but around 90% of the math in the IA needs to be HL math and more specifically math from the calculus section of the syllabus. You can also include math from outside the syllabus if needed but then you will need to explain that math is being used very clearly and precisely and why it is needed in the IA.
Each piece of math or any calculations you do have to be explained with clarity throughout the IA. Write the IA on a WORD document and use the equation section on word to properly layout the equations, working out and any math included.
Also be constantly referring to the IB Math IA criteria which I will provide you with so you know how and where marks are being awarded and what is needed to achieve the highest mark.
You may also need to obtain real life measurements on the museum which will need to be referenced properly and should be accurate in order to do the calculations.
Overall I want you to find the volume of the building, the surface area of the Arabic writing that is located all over and around the building and then finally the total surface area of the building using the IB Math AI HL syllabus and using mostly HL math.
Below is an overview of what the IA should include in order to gain high marks:
Overview of the Math IA The Math Internal Assessment (IA) is a unique opportunity for students to explore a mathematical topic of personal interest. For HL, it is essential to showcase mathematical rigor, creativity, and an ability to apply advanced concepts to real-world contexts. This IA will focus on exploring the architectural marvel of the Museum of the Future in Dubai through the lens of higher-level calculus, aiming to determine its surface area and volume. Specifics for Your IA
1. Introduction Begin with an introduction to the Museum of the Future, highlighting its unique toroidal (doughnut-like) shape and its status as a symbol of architectural and technological innovation. Clearly articulate the research focus: to explore and calculate the surface area and volume of the museum using higher-level calculus techniques from the IB Math AI HL syllabus. Explain your motivation for choosing this topic, such as a personal interest in architecture, mathematics, or design. Pose a guiding research question, such as: “How can higher-level calculus from the IB syllabus be applied to explore the surface area and volume of the Museum of the Future in Dubai?”
2. Mathematical Background Introduce the mathematical concepts that are likely to be relevant to the exploration, focusing on higher-level calculus from the IB syllabus. Discuss how mathematical modeling and integration are often used in architectural design and real-world problem-solving. Provide a brief overview of mathematical processes that might apply, such as: Representing 3D shapes through equations or curves. Using integrals to calculate volume and surface area.
3. Modeling the Museum of the Future Define the geometric features of the museum, emphasizing its toroidal design with curved surfaces and a hollow center. Discuss how you will model the shape mathematically, possibly using equations or graphical representations. State any assumptions or simplifications you plan to make about the museum’s structure to focus on the mathematical exploration (e.g., idealizing the shape as a smooth toroid).
4. Mathematical Exploration Dive into the main part of the IA, which involves: Modeling the structure mathematically, using higher-level calculus techniques. Applying methods from the syllabus to calculate surface area and volume. Allow space for flexibility in the methods chosen, as this is an open exploration. The goal is to use advanced calculus creatively and effectively to answer your research question. Include diagrams and visuals to help explain your process and reasoning.
5. Use of Technology Highlight how technology will play a crucial role in your exploration: Graphing tools (e.g., Desmos, GeoGebra) to visualize equations and shapes. Mathematical software (e.g., MATLAB, Python) for performing complex calculations and integrations. Discuss how these tools enhance the accuracy and depth of your exploration.
6. Reflection on Limitations Reflect on the challenges of modeling a complex architectural structure like the Museum of the Future. Acknowledge any simplifications or approximations made in your calculations and how they may impact the results. Suggest improvements or further areas of exploration, such as using more precise measurements or advanced software tools.
7. Applications and Relevance Discuss the broader implications of your findings, such as: How advanced mathematics can be applied to architecture and engineering. The relevance of surface area and volume calculations in sustainable design or material optimization. Reflect on how this exploration deepened your appreciation for the intersection of mathematics and the real world.
8. Conclusion Summarize your findings and the mathematical journey undertaken in the IA. Revisit your research question and explain how you addressed it through your exploration. Reflect on how this experience has enhanced your understanding of mathematics and its real-world applications

Consider an unbalanced study with six subjects, identified as A, B, C, D, E and G. In the actual study, • Subjects A and B are assigned to the first treatment, and the other subjects are assigned to the second treatment. • There are exactly two successes, obtained by A and C. This information is needed for parts (a)–(c) below. (a) Compute the observed value of the test statistic. (b) Assume that the Skeptic is correct. Determine the observed value of the test statistic for the assignment that places D and E on the first treatment, and the remaining subjects on the second treatment. (c) We have obtained the sampling distribution of the test statistic on the assumption that the Skeptic is correct. It also is possible to obtain a sampling distribution of the test statistic if the Skeptic is wrong provided we specify exactly how the Skeptic is in error. These new sampling distributions are used in the study of statistical power which is briefly described in Chapter 7 of the text. Assume that the Skeptic is correct about subjects A and G, but incorrect about subjects B, C, D and E. For the assignment that puts D and G on the first treatment, and the other subjects on the second treatment, determine the response for each of the six subjects.

A sample of size 40 yields the following sorted data. Note that I have x-ed out x(39) (the second largest number).
This fact will NOT prevent you from answering the questions below. 14.1 46.0 49.3 53.0 54.2 54.7 54.7 54.7 54.8 55.4 57.6 58.2 58.3 58.7 58.9 60.8 60.9 61.0 61.1 63.0 64.3 65.6 66.3 66.6 67.0 67.9 70.1 70.3 72.1 72.4 72.9 73.5 74.2 75.3 75.4 75.9 76.5 77.0 x 88
(a) Calculate range, IQR, and median of these data
. (b) Given that the mean of these data is 63.50 (exactly) and the standard deviation is 12.33, what proportion of the data lie within one standard deviation of the mean?

1 )Outliers & Missing Data: Answer the following questions:
a. True or False: Non-random missing data are often times more problematic than random missing data.
b. True or False: A normal practice of dealing with missing data is to eliminate them regardless of the amount as they can introduce severe bias into the results. 
c. Use Outlier.sav and identify potential outliers. Provide the reasons of such identification. [Hint: A boxplot might help]
d. Given the sample size in Outlier.sav, what seems to be the most appropriate remedy for the outliers?
3 Scholarly references

Starting from an airport, an airplane flies 225 miles northwest, then 150 miles south-west.
Answer the following:
Draw a graph or figure to represent this situation.
Describe how the concepts of vectors and complex plane can be applied in this case.
How far, in miles, from the airport is the plane?
Provide another example of a scenario that involves the same concept.
Note: When drawing the diagram, try to be as consistent as possible.
That is, when drawing the vector for length 225 miles, make this line about 1.5 times longer than the vector representing 150 miles.
When there is a reference to “northwest” this means to draw a vector to the left of north, where the angle between the vector and the vertical axis is 45 degrees.

Starting from an airport, an airplane flies 225 miles northwest, then 150 miles south-west.
Answer the following:
Draw a graph or figure to represent this situation.
Describe how the concepts of vectors and complex plane can be applied in this case.
How far, in miles, from the airport is the plane?
Provide another example of a scenario that involves the same concept.
Note: When drawing the diagram, try to be as consistent as possible.
That is, when drawing the vector for length 225 miles, make this line about 1.5 times longer than the vector representing 150 miles.
When there is a reference to “northwest” this means to draw a vector to the left of north, where the angle between the vector and the vertical axis is 45 degrees.