J.P. McCarthy: Math Page

Another Relative Velocity Question

December 12, 2018 in Leaving Cert Applied Maths | Leave a comment

On a particular day the velocity of the wind, in terms of $\mathbf{i}$ and $\mathbf{j}$ , is $x\mathbf{i}-3\mathbf{j}$ , where $x\in\mathbb{N}$ .

$\mathbf{i}$ and $\mathbf{j}$ are unit vectors in the directions East and North respectively.

To a man travelling due East the wind appears to come from a direction North $\alpha^\circ$ West where $\tan\alpha=2$ .

When he travels due North at the same speed as before, the wind appears to come from a direction North $\beta^\circ$ West where $\tan\beta=3/2$ .

Find the actual direction of the wind.

Solution:

We start by writing

$\overrightarrow{V_{WM}}=\overrightarrow{V_W}-\overrightarrow{V_M}$ .

We have two equations.

Firstly, when the man travels due East, $\overrightarrow{V_M}=a\mathbf{i}$ for some constant $a$ . We have $\overrightarrow{V_W}=x\mathbf{i}-3\mathbf{j}$ and so

$\overrightarrow{V_{WM}}=(x-a)\mathbf{i}-3\mathbf{j}$ .

We know that this, with respect to the man, is coming from North $\alpha^\circ$ West. This means we have:

And furthermore:

$\displaystyle \tan\alpha=\frac{x-a}{3}\overset{!}{=}2$

$\Rightarrow x-a=6\Rightarrow a=x-6$ .

Now consider when the man travels due North, $\overrightarrow{V_M}=a\mathbf{j}$ . We have $\overrightarrow{V_W}=x\mathbf{i}-3\mathbf{j}$ still and so

$\overrightarrow{V_{WM}}=x\mathbf{i}-(3+a)\mathbf{j}$ .

We know that this, with respect to the man, is coming from North $\beta^\circ$ West. This means we have:

And furthermore:

$\displaystyle \tan\beta=\frac{x}{3+a}\overset{!}{=}\frac32$

$\underset{\times_{2(3+a)}}{\Rightarrow} 2x=9+3a$ ,

but we have $a=x-6$ and so

$2x=9+3(x-6)\Rightarrow x=9$ ,

so that $\overrightarrow{V_W}=9\mathbf{i}-3\mathbf{j}$ .

This means that the velocity of the wind looks like:

That is the wind comes from North $\theta$ West. We have that $\tan\theta=9/3\underset{\tan^{-1}}{\Rightarrow} \theta=\tan^{-1}(3)\approx 71.57^\circ$ , so the answer to the question asked is N $71.57^\circ$ W.

Relative Velocity Question Three Ways

December 12, 2018 in Leaving Cert Applied Maths | Leave a comment

Here we present three solutions to the one problem. The vector solution is probably the slickest. The geometry solution here can be simplified by being less rigorous, and the coordinate geometry solution might be made easier by using the $-b\pm\sqrt{\cdots}$ formula.

LCHL 2007, Q. 2(a)

Ship B is travelling west at 24 km/h. Ship A is travelling north at 32 km/h.
At a certain instant ship B is 8 km north-east of ship A.

(i) Find the velocity of ship A relative to ship B.

(ii) Calculate the length of time, to the nearest minute, for which the ships
are less than or equal to 8 km apart.

Solution to (i): We have that $\overrightarrow{V_A}=32\mathbf{j}$ and $\overrightarrow{V_B}=-24\mathbf{i}$ and so

$\overrightarrow{V_{AB}}=\overrightarrow{V_A}-\overrightarrow{V_B}=32\mathbf{j} -(-24\mathbf{i})=24\mathbf{i}+32\mathbf{j}$ .

Vector Approach to (ii)

First of all we draw a picture. As we are talking relative to ship B we will put B at the origin. If ship B is 8 km north-east of ship A then ship A is 8 km south-west of ship B.

rel1

Where $\overrightarrow{R_0}$ is the initial displacement of ship A relative to ship B, the displacement of ship A relative to ship B, as a function of time, is given by

$\overrightarrow{R_{AB}}(t)=\overrightarrow{R_0}+t\cdot \overrightarrow{V_{AB}}$ .

Using some trigonometry — vertical and horizontal components of $\overrightarrow{R_0}$ — we have

$\displaystyle\overrightarrow{R_0}=-\frac{8}{\sqrt{2}}\mathbf{i}-\frac{8}{\sqrt{2}}\mathbf{j}$ ,

and so

$\displaystyle \overrightarrow{R_{AB}(t)}=\left(24t-\frac{8}{\sqrt{2}}\right)\mathbf{i}+\left(32t-\frac{8}{\sqrt{2}}\right)\mathbf{j}$ .

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MATH6055: Winter 2018, Week 12

December 6, 2018 in MATH6055 | Leave a comment

Transposition Survey

If you would like to help us more with the Transposition Project, please take this survey.

Mathematics Exam Advice

The first piece of advice is to read questions carefully. Don’t glance at a question and go off writing: take a moment to understand what you have been asked to do.
Don’t use tippex; instead draw a simple line(s) through work that you think is incorrect.
For equations, check your solution by substituting your solution into the original equation. If your answer is wrong and you know it is wrong: write that on your script.

If you do have time at the end of the exam, go through each of your answers and ask yourself:

have I answered the question that was asked?
does my answer make sense? If no, say so, and then try and fix your solution.
check your answer (e.g. if you are looking at a general true, look at a special case; substitute your solution into equations; check your answer against a rough estimate; or what a picture is telling you; etc). If your answer is wrong, say so, and then try and fix your solution.

Student Feedback

You are invited to give your feedback on my teaching and this module here.

CA Results

Have been emailed to you.

Sample Papers

I have not one but two sample exams for ye.

We will go through MATH6055 Sample 1 in Week 13 (I will give you a paper copy of this on Monday) and here is an additional Sample Exam 2 for you.

Both Sample Exams should be considered under the following understanding:

This sample has been drafted to give you an idea of the NEW MATH6055 LAYOUT and is no indication of the specifics, difficulty or length of any individual questions.

Week 12

We finished our study of Graph Theory by looking at connectedness, degree, walks (Monday), and trees. Eulerian graphs (Tuesday), Hamiltonian graphs, and Dirac’s Theorem (Friday).

Week 13

We will have three review lectures, and tutorials as normal.

I have drafted a sample paper and we will go through this exam on the board and then I will answer your questions if there are any. If there are none I will help one-to-one. Usual class times and locations.

Tutorials on Week 13 at the usual times and venue. As long as there are not too many people in a tutorial (max 18), you can attend both tutorials if you want.

Study

Please feel free to ask me questions about the exercises via email or even better on this webpage — especially those of us who struggled in the test.

Student Resources

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, exam papers etc.

MATH7019: Winter 2018, Week 12

December 6, 2018 in MATH7019 | Leave a comment

Mathematics Exam Advice

The first piece of advice is to read questions carefully. Don’t glance at a question and go off writing: take a moment to understand what you have been asked to do.
Don’t use tippex; instead draw a simple line(s) through work that you think is incorrect.
For equations, check your solution by substituting your solution into the original equation. If your answer is wrong and you know it is wrong: write that on your script.

If you do have time at the end of the exam, go through each of your answers and ask yourself:

have I answered the question that was asked?
does my answer make sense? If no, say so, and then try and fix your solution.
check your answer (e.g. for a fitted curve or beam function, input values and see do they make sense; substitute your solution into equations; check your answer against a rough estimate; or what a picture is telling you; etc). If your answer is wrong, say so, and then try and fix your solution.

Student Feedback

You are invited to give your feedback on my teaching and this module here.

Assessment 2

I am working my way through these. I promise you your final CA results before Monday morning. Some comments on Assignment 2.

Week 12

We finished off Chapter 4 by looking at Error Analysis, including rounding error. We had a tutorial for a lecture on Wednesday, and Thursday might allow some tutorial time.

Week 13

We will go through last year’s exam on the board and then I will answer your questions if there are any. If there are none I will help one-to-one. Usual class times and locations.

We will also have tutorials on Friday 14 December at the usual times and venue. If there aren’t too many students present (max 18), you can attend both tutorials.

Past exam papers (Winter and Autumn) may be found here.

Study

Please feel free to ask me questions about the exercises via email or even better on this webpage — especially those of us who struggled in the test.

Student Resources

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, past exam papers, etc.

MATH6040: Winter 2018, Week 12

December 6, 2018 in MATH6040 | Leave a comment

Mathematics Exam Advice

The first piece of advice is to read questions carefully. Don’t glance at a question and go off writing: take a moment to understand what you have been asked to do.
Don’t use tippex; instead draw a simple line(s) through work that you think is incorrect.
For equations, check your solution by substituting your solution into the original equation. If your answer is wrong and you know it is wrong: write that on your script.

If you do have time at the end of the exam, go through each of your answers and ask yourself:

have I answered the question that was asked?
does my answer make sense?
check your answer (e.g. differentiate/antidifferentiate an antiderivative/derivative, substitute your solution into equations, check your answer against a rough estimate, or what a picture is telling you, etc)

Student Feedback

You are invited to give your feedback on my teaching and this module here.

CA Results

Have been emailed out, with comments.

Week 12

On Monday and Tuesday we had extra tutorials: you were invited to either work on integration and/or matrices; the choice was up to you.

On Thursday we finished off the module by doing an extra example of a centre of gravity of a solid of revolution. We had tutorial time after that.

Week 13

There is an exam paper at the back of your notes — I will go through this on the board in the lecture times (in the usual venues):

Monday 16:00
Tuesday 09:00
Thursday 09:00

We will also have tutorial time in the tutorial slots. You can come to as many tutorials as you like.

Monday at 09:00 in E15
Monday at 17:00 in B189
Thursday at 12:00 in E4

Past exam papers (MATH6040 runs in Semester 1 and Semester 2) may be found here.

Recall that this module is MATH6040: Technological Maths 201 and not MATH6015: Technological Maths 2.

Study

Please feel free to ask me questions about the exercises via email or even better on this webpage.

Student Resources

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, past exam papers, etc.

MATH6055: Winter 2018, Week 11

November 29, 2018 in MATH6055 | Leave a comment

Student Feedback

You are invited to give your feedback on my teaching and this module here.

Test 2

I hope to get the Test 2 results, your collated CA results, and the Test 2 Marking Scheme out to you by the end of next week.

Sample Papers

I have not one but two sample exams for ye.

We will go through MATH6055 Sample in Week 13 (I will give you a paper copy of this on Monday) and here is an additional Sample Exam for you.

Both Sample Exams should be considered under the following understanding:

This sample has been drafted to give you an idea of the NEW MATH6055 LAYOUT and is no indication of the specifics, difficulty or length of any individual questions.

Week 11

In Week 11 we explored Network Theory, or rather Graph Theory, in more depth. In particular we put them on a firm set-theoretic foundation.

Test 2 was on Friday.

Week 12

We will finish our study of Graph Theory by looking at connectedness, degree, walks, and trees. Eulerian graphs, Fleury’s Algorithm, Hamiltonian graphs, and Dirac’s Theorem.

Week 13

We will have three review lectures, and tutorials as normal.

Tutorials on Week 13 at the usual times and venue.

Study

Please feel free to ask me questions about the exercises via email or even better on this webpage — especially those of us who struggled in the test.

Student Resources

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc.

MATH7019: Winter 2018, Week 11

November 29, 2018 in MATH7019 | Leave a comment

Student Feedback

You are invited to give your feedback on my teaching and this module here.

Assessment 2

I have started correcting these: I won’t make any promises but will do my utmost to have them to you as soon as possible, certainly sooner than the end of next week.

Week 11

We looked at more general Taylor Series: not just near $a=0$ — and also for functions of several variables. Here are some reasons why an engineer might be interested in Taylor Series of functions of several variables.

Week 12

We will finish off Chapter 4 by looking at Error Analysis, including rounding error. We should be able to hold an additional tutorial or two.

Week 13

We will go through last year’s exam on the board and then I will answer your questions if there are any. If there are none I will help one-to-one. Usual class times and locations.

We will also have tutorials on Friday 14 December at the usual times and venue.

Study

Please feel free to ask me questions about the exercises via email or even better on this webpage — especially those of us who struggled in the test.

Student Resources

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc.

MATH6040: Winter 2018, Week 11

November 29, 2018 in MATH6040 | Leave a comment

Student Feedback

You are invited to give your feedback on my teaching and this module here.

Test 2 Results

Will be sent out tomorrow afternoon along with some comments.

Week 11

We had Test 2 on Monday.

On Tuesday we had a tutorial, doing some revision exercises for antidifferentiation and integration (Q. 5 should be +400 J, Q. 9 should be 3e-3).

On Thursday we looked at centres of gravity of solids of revolution.

Week 12

On Monday and Tuesday we will have extra tutorials: you will be invited to either work on integration and/or matrices; the choice will be up to you.

On Thursday we will finish off the module by doing an extra example of a centre of gravity of a solid of revolution.

Week 13

There is an exam paper at the back of your notes — I will go through this on the board in the lecture times (in the usual venues):

Monday 16:00
Tuesday 09:00
Thursday 09:00

We will also have tutorial time in the tutorial slots. You can come to as many tutorials as you like.

Monday at 09:00 in E15
Monday at 17:00 in B189
Thursday at 12:00 in E4

Study

Please feel free to ask me questions about the exercises via email or even better on this webpage.

Student Resources

Please see the Student Resources tab on the top of this page for information on past exam papers, Academic Learning Centre, etc.

Expected Number of Shared Birthdays

November 28, 2018 in General | Leave a comment

I received the following email (extract) from a colleague:

With the birthday question the chances of 23 people having unique birthdays is less than ½ so probability of shared birthdays is greater than 1-in-2.

Coincidentally on the day you sent out the paper, the following question/math fact was in my son’s 5th Class homework.

We are still debating the answer, hopefully you could clarify…

In a group of 368 people, how many should share the same birthday. There are 16×23 in 368 so there are 16 ways that 2 people should share same birthday (?) but my son pointed out, what about 3 people or 4 people etc.

I don’t think this is an easy problem at all.

First off we assume nobody is born on a leap day and the distribution of birthdays is uniform among the 365 possible birthdays. We also assume the birthdays are independent (so no twins and such).

They were probably going for 16 or 32 but that is wrong both for the reasons given by your son but also for the fact that people in different sets of 23 can also share birthdays.

The brute force way of calculating it is to call by $X$ the random variable that is the number of people who share a birthday and then the question is more or less looking for the expected value of $X$ , which is given by:

$\displaystyle \mathbb{E}[X]=\sum_{i=2}^{368}i\cdot \mathbb{P}[X=i]$ .

Already we have that $\mathbb{P}[X=2]=\mathbb{P}[X=3]=0$ (why), and $\mathbb{P}[X=4]$ is (why) the probability that four people share one birthday and 364 have different birthdays. This probability isn’t too difficult to calculate (its about $10^{-165}$ ) but then things get a lot harder.

For $X=5$ , there are two possibilities:

5 share a birthday, 363 different birthdays, OR
2 share a birthday, 3 share a different birthday, and the remaining 363 have different birthdays

Then $X=6$ is already getting very complex:

6 share a birthday, 362 different birthdays, OR
3, 3, 362
4, 2, 362
2, 2, 2, 362

This problem is spiraling out of control.

There is another approach that takes advantage of the fact that expectation is linear, and the probability of an event $E$ not happening is

$\displaystyle\mathbb{P}[\text{not-}E]=1-\mathbb{P}[E]$ .

Label the 368 people by $i=1,\dots,368$ and define a random variable $S_i$ by

$\displaystyle S_i=\left\{\begin{array}{cc}1&\text{ if person i shares a birthday with someone else} \\ 0 & \text{ if person i does not share a birthday}\end{array}\right.$

Then $X$ , the number of people who share a birthday, is given by:

$\displaystyle X=S_1+S_2+\cdots+S_{368}$ ,

and we can calculate, using the linearity of expectation.

$\mathbb{E}[X]=\mathbb{E}[S_1]+\cdots \mathbb{E}[S_{368}]$ .

The $S_i$ are not independent but the linearity of expectation holds even when the addend random variables are not independent… and each of the $S_i$ has the same expectation. Let $p$ be the probability that person $i$ does not share a birthday with anyone else; then

$\displaystyle\mathbb{E}[S_i]=0\times\mathbb{P}[S_i=0]+1\times \mathbb{P}[S_i=1]$ ,

but $\displaystyle\mathbb{P}[S_i=0]=\mathbb{P}[\text{ person i does not share a birthday}]=p$ , and

$\displaystyle \mathbb{P}[S_i=1]=\mathbb{P}[\text{not-}(S_i=0)]=1-\mathbb{P}[S_i=0]=1-p$ ,

and so

$\displaystyle\mathbb{E}[S_i]=1-p$ .

All of the 368 $S_i$ have this same expectation and so

$\displaystyle\mathbb{E}[X]=368\cdot (1-p)$ .

Now, what is the probability that nobody shares person $i$ ‘s birthday?

We need persons $1\rightarrow i-1$ and $i+1\rightarrow 368$ — 367 persons — to have different birthdays to person $i$ , and for each there is 364/365 ways of this happening, and we do have independence here (person 1 not sharing person $i$ ‘s birthday doesn’t change the probability of person 2 not sharing person $i$ ‘s birthday), and so $\mathbb{P}[\text{(person k not sharing) AND (person k not sharing)}]$ is the product of the probabilities.

So we have that

$\displaystyle p=\left(\frac{364}{365}\right)^{367}$ ,

and so the answer to the question is:

$\displaystyle\mathbb{E}[X]=368\cdot \left(1-\left(\frac{364}{365}\right)^{367}\right)\approx 233.54\approx 234$ .

There is possibly another way of answering this using the fact that with 368 people there are

$\displaystyle \binom{368}{2}=67528$

pairs of people.

MATH6055: Winter 2018, Week 10

November 23, 2018 in MATH6055 | Leave a comment

Transposition Project – Part II

October 22 you took a quiz as part of the Transposition Project that the Mathematics Department is undertaking in an effort to improve our teaching.

You will have another 15 minute quiz on Monday.

If you do not have an internet ready device, or did not do the first quiz, you may leave class early.

Thank you again for your participation.

Test 2

Test 2, worth 15% of your final grade, based on Chapter 3: Algebra, will take place on Friday 30 November in the usual lecture venue of B214.

All Chapter 3 material is examinable, but the test will be similar to the sample test.

There will also be a question on the growth rate of functions: the relevant info to answer these questions are on p. 81, and they are exercises 18, 19 on p. 79.

Here is a sample test.

The sample is to give you an idea of the length of the test. You know from Test 1 the layout (i.e. you write your answers on the paper). You will be allowed use a calculator for all questions.

I strongly advise you that, for those who might have done poorly, or not particularly well, in Test 1, attending tutorials alone will not be sufficient preparation for this test, and you will have to devote extra time outside classes to study aka do exercises.

Week 10

We introduced and studied the properties of uses of logarithms. We began the chapter on Network Theory by looking at the Bridges of Konigsberg Problem.

Week 11

In Week 11 we will explore Network Theory, or rather Graph Theory, in more depth. We will look at digraphs, connectedness, degree, walks, and trees.

Test 2 will be on Friday.

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J.P. McCarthy: Math Page

Solution:

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LCHL 2007, Q. 2(a)

Vector Approach to (ii)

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Transposition Survey

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CA Results

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Mathematics Exam Advice

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Transposition Project – Part II

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