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Sphere packing problem: Pyramid design

Sphere packing problems are a maths problems which have been considered over many centuries – they concern the optimal way of packing spheres so that the wasted space is minimised.  You can achieve an average packing density of around 74% when you stack many spheres together, but today I want to explore the packing density of 4 spheres (pictured above) enclosed in a pyramid.

Considering 2 dimensions

 

First I’m going to consider the 2D cross section of the base 3 spheres.  Each sphere will have a radius of 1.  I will choose A so that it is at the origin.  Using some basic Pythagoras this will give the following coordinates:

Finding the centre

 

Next I will stack my single sphere on top of these 3, with the centre of this sphere directly in the middle.  Therefore I need to find the coordinate of D.  I can use the fact that ABC is an equilateral triangle and so:

3D coordinates

Next I can convert my 2D coordinates into 3D coordinates.  I define the centre of the 3 base circles to have 0 height, therefore I can add z coordinates of 0.  E will be the coordinate point with the same x and y coordinates as D, but with a height, a, which I don’t yet know:

In order to find I do a quick sketch, seen below:

Here I can see that I can find the length AD using trig, and then the height DE (which is my a value) using Pythagoras:

Drawing spheres

The general equation for spheres with centre coordinate (a,b,c) and radius 1 is:

Therefore the equation of my spheres are:

Plotting these on Geogebra gives:

 

 

Drawing a pyramid

Next I want to try to draw a pyramid such that it encloses the spheres.  This is quite difficult to do algebraically – so I’ll use some technology and a bit of trial and error.

First I look at creating a base for my pyramid.  I’ll try and construct an equilateral triangle which is a tangent to the spheres:

This gives me an equilateral triangle with lengths 5.54. I can then find the coordinate points of F,G,H and plot them in 3D.  I’ll choose point E so that it remains in the middle of the shape, and also has a height of 5.54 from the base. This gives the following:

As we can see, this pyramid does not enclose the spheres fully.  So, let’s try again, this time making the base a little bit larger than the 3 spheres:

 

 

This gives me an equilateral triangle with lengths 6.6.  Taking the height of the pyramid to also be 6.6 gives the following shape:

 

This time we can see that it fully encloses the spheres.  So, let’s find the density of this packing.  We have:

Therefore this gives:

and we also have:

Therefore the density of our packaging is:

Given our diagram this looks about right – we are only filling less than half of the available volume with our spheres.

Comparison with real data

 

[Source: Minimizing the object dimensions in circle and sphere packing problems]

We can see that this task has been attempted before using computational power – the table above shows the average density for a variety of 2D and 3D shapes.  The pyramid here was found to have a density of 46% – so our result of 44% looks pretty close to what we should be able to achieve.  We could tweak our measurements to see if we could improve this density.

So, a nice mixture of geometry, graphical software, and trial and error gives us a nice result.  You could explore the densities for other 2D and 3D shapes and see how close you get to the results in the table.

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The Tusi couple – A circle rolling inside a circle

Numberphile have done a nice video where they discuss some beautiful examples of trigonometry and circular motion and where they present the result shown above: a circle rolling within a circle, with the individual points on the small circle showing linear motion along the diameters of the large circle.  So let’s see what maths we need to create the image above.

 Projection of points

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We can start with the equation of a unit circle centred at the origin:

Screen Shot 2020-06-18 at 7.01.45 PM

and we can then define a point on this circle parametrically by the coordinate:

Screen Shot 2020-06-18 at 7.01.49 PM

Here t is the angle measured from the horizontal.

If we then want to see the projection of this point along the y-axis we can also plot:

Screen Shot 2020-06-18 at 7.01.53 PM

and to see the projection of this point along the x-axis we can also plot:

Screen Shot 2020-06-18 at 7.02.06 PM

By then varying t from 0 to 2 pi gives the animation above – where the black dot on the circle moves around the circle and there is a projection of its x and y coordinates on the axes.

Projection along angled lines

Screen Shot 2020-06-18 at 7.32.46 PM

I can then add a line through the origin at angle a to the horizontal:

Screen Shot 2020-06-18 at 7.35.25 PM

and this time I can project so that the line joining up the black point on the edge of the large circle intersects the dotted line in a right angle.

In order to find the parametric coordinate of this point projection I can use some right angled triangles as follows:

Screen Shot 2020-06-18 at 7.57.20 PM

The angle from the horizontal to my point A is t.  The angle from the horizontal to the slanted line is a.  The length of my radius BA is 1.  This gives me the length of BC.

Screen Shot 2020-06-18 at 7.53.30 PM

But I have the identity:

Screen Shot 2020-06-18 at 7.53.34 PM

Therefore this gives:

Screen Shot 2020-06-18 at 7.53.37 PM

And using some more basic trigonometry gives the following diagram:

Screen Shot 2020-06-18 at 8.00.53 PM

 

Therefore the parametric form of the projection of the point can be given as:

Screen Shot 2020-06-18 at 7.40.17 PM

Adding more lines

Screen Shot 2020-06-18 at 8.03.32 PM

I can add several more slanted lines through the origin.  You can see that each dot on the line is the right angle projection between the line and the point on the circle.  As we do this we can notice that the points on the lines look as though they form a circle.  By noticing that the new smaller circle is half the size of the larger circle, and that the centre of the smaller circle is half-way between the origin and the point on the large circle, we get:

Screen Shot 2020-06-18 at 8.07.58 PM

Screen Shot 2020-06-18 at 8.10.16 PM

We can the vary the position of the point on the large circle to then create our final image:

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We have a connection between both linear motion and circular motion and create the impression of a circle rolling inside another.

You can play around with this demos graph here.  All you need to do is either drag the black point around the circle, or press play for the t slider.

More ideas on projective geometry:

Screen Shot 2020-06-18 at 8.16.30 PM

Ferenc Beleznay has made this nice geogebra file here which shows a different way of drawing a connection between a moving point on a large circle and a circle half the size. Here we connect the red dot with the origin and draw the perpendicular from this line to  the other edge of the small circle.  The point of intersection of the two lines is always on the small circle.

Further exploration 

There is a lot more you can explore – start by looking into the Tusi Couple – which is what we have just drawn – and the more general case the hypocycloid.

 

Screen Shot 2019-02-20 at 2.24.37 PM

Modeling hours of daylight

Desmos has a nice student activity (on teacher.desmos.com) modeling the number of hours of daylight in Florida versus Alaska – which both produce a nice sine curve when plotted on a graph.  So let’s see if this relationship also holds between Phuket and Manchester.

First we can find the daylight hours from this site, making sure to convert the times given to decimals of hours.

Phuket

Phuket has the following distribution of hours of daylight (taking the reading from the first of each month and setting 1 as January)

Screen Shot 2019-02-20 at 2.08.34 PM

Manchester 

Manchester has much greater variation and is as follows:

Screen Shot 2019-02-20 at 2.08.40 PM

Therefore when we plot them together (Phuket in green and Manchester in blue) we get the following 2 curves:

Screen Shot 2019-02-20 at 2.09.00 PM

We can see that these very closely fit sine curves, indeed we can see that the following regression lines fit the curves very closely:

Manchester:

Screen Shot 2019-02-20 at 2.08.54 PM

Phuket:

Screen Shot 2019-02-20 at 2.08.46 PM

For Manchester I needed to set the value of b (see what happens if you don’t do this!) Because we are working with Sine graphs, the value of d will give the equation of the axis of symmetry of the graph, which will also be the average hours of daylight over the year.  We can see therefore that even though there is a huge variation between the hours of daylight in the 2 places, they both get on average the same amount of daylight across the year (12.3 hours versus 12.1 hours).

Further investigation:

Does the relationship still hold when looking at hours of sunshine rather than daylight?  How many years would we expect our model be accurate for?  It’s possible to investigate the use of sine waves to model a large amount of natural phenomena such as tide heights and musical notes – so it’s also possible to investigate in this direction as well.

IB Revision

If you’re already thinking about your coursework then it’s probably also time to start planning some revision, either for the end of Year 12 school exams or Year 13 final exams. There’s a really great website that I would strongly recommend students use – you choose your subject (HL/SL/Studies if your exam is in 2020 or Applications/Analysis if your exam is in 2021), and then have the following resources:

Screen Shot 2018-03-19 at 4.42.05 PM.pngThe Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and each area then has a number of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

Screen Shot 2019-07-27 at 10.02.40 AM

The Practice Exams section takes you to ready made exams on each topic – again with worked solutions.  This also has some harder exams for those students aiming for 6s and 7s and the Past IB Exams section takes you to full video worked solutions to every question on every past paper – and you can also get a prediction exam for the upcoming year.

I would really recommend everyone making use of this – there is a mixture of a lot of free content as well as premium content so have a look and see what you think.

Screen Shot 2018-03-25 at 5.01.18 PM.png

Modelling tides: What is the effect of a full moon?

Let’s have a look at the effect of the moon on the tides in Phuket.  The Phuket tide table above shows the height of the tide (meters) on given days in March, with the hours along the top.  So if we choose March 1st (full moon) we get the following graph:

Phuket tide at full moon:

Screen Shot 2018-03-25 at 5.07.54 PM

If I use the standard sine regression on Desmos I get the following:

Screen Shot 2018-03-25 at 5.12.55 PM

Screen Shot 2018-03-25 at 5.13.00 PM

This doesn’t look a very useful graph – but the R squared value is very close to one – so what’s gone wrong?  Well, Desmos has done what we asked it to do – found a sine curve that goes through the points, it’s just that it’s chosen a b value of close to 120 – meaning that the curve has a very small period.  So to prevent Desmos doing this, we need to fix the period first.   If we are in radians the we use the formula period = 2pi / b.  Therefore looking at the original graph we can see that this period is around 12.  Therefore we have:

period = 2pi/b

12 = 2pi/b

b = 2pi/12 or pi/6.

Plotting this new graph gives something that looks a lot nicer:

Screen Shot 2018-03-25 at 5.17.48 PM

Screen Shot 2018-03-25 at 5.17.53 PM

Phuket tide at new moon:

Screen Shot 2018-03-25 at 5.26.45 PM

Screen Shot 2018-03-25 at 5.26.40 PM

Analysis:

Both graphs show a very close fit to the original data – though both under-value the tide at 2300.  We can see that the full moon has indeed had an effect on the amplitude of the sine curves – with the amplitude of 1.21m for the full moon and only 1.03m for the new moon.

Further study:

We could then see if this relationship holds throughout the year – is there a general formula to explain the moons effect on the amplitude?  We could also see how we have to modify the sine wave to capture the tidal height over an entire week or month.  Can we capture it with a single equation (perhaps a damped sine wave?) or is it only possible as a piecewise function?  We could also use some calculus to find the maximum and minimum points.

There is a very nice pdf which goes into more detail on the maths behind modeling tides here.  There we go – a nice simple investigation which can be expanded in a number of directions.

IB Revision

Screen Shot 2018-03-19 at 4.35.19 PM

If you’re already thinking about your coursework then it’s probably also time to start planning some revision, either for the end of Year 12 school exams or Year 13 final exams. There’s a really great website that I would strongly recommend students use – you choose your subject (HL/SL/Studies if your exam is in 2020 or Applications/Analysis if your exam is in 2021), and then have the following resources:

Screen Shot 2018-03-19 at 4.42.05 PM.pngThe Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and each area then has a number of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

Screen Shot 2019-07-27 at 10.02.40 AM

The Practice Exams section takes you to ready made exams on each topic – again with worked solutions.  This also has some harder exams for those students aiming for 6s and 7s and the Past IB Exams section takes you to full video worked solutions to every question on every past paper – and you can also get a prediction exam for the upcoming year.

I would really recommend everyone making use of this – there is a mixture of a lot of free content as well as premium content so have a look and see what you think.

Circular Motion: Modelling a ferris wheel

This is a nice simple example of how the Tracker software can be used to demonstrate the circular motion of a Ferris wheel.  This is sometimes asked in IB maths exams – so it’s nice to get a visual representation of what is happening.

First I took a video from youtube of a Ferris wheel, loaded it into Tracker, and then used the program to track the position of a single carriage as it moved around the circle.  I then used Tracker’s graphing capabilities to plot the height of the carriage (y) against time (t).  This produces the following graph:

As we can see this is a pretty good fit for a sine curve. So let’s use the regression tool to find what curve fits this:

The pink curve with the equation:

y = -116.1sin(0.6718t+2.19)

fits reasonably well.  If we had the original dimensions of the wheel we could scale this so the y scale represented the metres off the ground of the carriage.

There we go!  Short and simple, but a nice starting point for an investigation on circular motion.

IB Revision

Screen Shot 2018-03-19 at 4.35.19 PM

If you’re already thinking about your coursework then it’s probably also time to start planning some revision, either for the end of Year 12 school exams or Year 13 final exams. There’s a really great website that I would strongly recommend students use – you choose your subject (HL/SL/Studies if your exam is in 2020 or Applications/Analysis if your exam is in 2021), and then have the following resources:

Screen Shot 2018-03-19 at 4.42.05 PM.pngThe Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and each area then has a number of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

Screen Shot 2019-07-27 at 10.02.40 AM

The Practice Exams section takes you to ready made exams on each topic – again with worked solutions.  This also has some harder exams for those students aiming for 6s and 7s and the Past IB Exams section takes you to full video worked solutions to every question on every past paper – and you can also get a prediction exam for the upcoming year.

I would really recommend everyone making use of this – there is a mixture of a lot of free content as well as premium content so have a look and see what you think.

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IB Maths Exploration Guide

IB Maths Exploration Guide

A comprehensive 63 page pdf guide to help you get excellent marks on your maths investigation. Includes:

  1. Investigation essentials,
  2. Marking criteria guidance,
  3. 70 hand picked interesting topics
  4. Useful websites for use in the exploration,
  5. A student checklist for top marks
  6. Avoiding common student mistakes
  7. A selection of detailed exploration ideas
  8. Advice on using Geogebra, Desmos and Tracker.

Available to download here.

IB Exploration Modelling and Statistics Guide


IB Exploration Modelling and Statistics Guide

A 60 page pdf guide full of advice to help with modelling and statistics explorations – focusing in on non-calculator methods in order to show good understanding. Includes:

  1. Pearson’s Product: Height and arm span
  2. How to calculate standard deviation by hand
  3. Binomial investigation: ESP powers
  4. Paired t tests and 2 sample t tests: Reaction times
  5. Chi Squared: Efficiency of vaccines
  6. Spearman’s rank: Taste preference of cola
  7. Linear regression and log linearization.
  8. Quadratic regression and cubic regression.
  9. Exponential and trigonometric regression.

Available to download here.

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