[Pri20150522RPRW] More Hands make Light Work?

Question

Introduction

     The key to solving this is to find the total rate of work for Ahmad, Kumar and Calvin.  There is a relationship Work = Rate ´ Time, similar to the relationship Distance = Speed ´ Time, and you can use a similar triangle mnemonic for it.  For example if you cover ‘R’ with your finger, you get the relation Rate = Work / Time.  To organise your information, you may use a table to tabulate the given data.

     Do you believe “more hands make light work”?  Or is it “too many cooks spoil the broth”?  In real life, people may not work harmoniously together, or work at the same rate (never getting tired).  They may get distracted by Facebook, mobile phones or office gossip.  In school mathematics problems, we assume that when people work together, we can just add up their rates of work.  Yes, it is quite funny, but let us just assume.

Solution

Ans: 10 days

Commentary

     When we add up the given rates, we get  ¹/₅,  which is the rate that 2 Ahmads, 2 Kumars and 2 Calvins would work.  Unfortunately we do not have the clones.  We just have one Ahmad, one Kuman and one Calvin.  So we divide that by 2 to get  ¹/₁₀  and this is their combined rate.  This means that they can complete  1  house in  10 days.

     It is possble to do this question without the table, for example by taking the LCM of the denominators and considering how many houses can the guys build in  60  days.  However, the table is a good way of organising information and you can solve the problem by considering the total rate.

HeuristicsUsed

H02. Use a diagram / model   (including table)

H03. Make a systematic list

H05. Work backwards

H09. Restate the problem in another way

H11. Solve part of the problem

Suitable Levels

* Primary School Mathematics

* other syllabuses that involve fractions and ratios

[EM_20150412SCA] Is your Airport Design career “taking off”?

Question

An Airbus 380 has constant acceleration of  1 m/s2.  Its takeoff velocity is 280 km/h.  How long must the runway be at a minimum to allow the plane to take off?

Introduction

     A practical question for airport design, perhaps?  I present two solutions.  The first uses a graphical method (speed-time graph) which is in the (“Elementary”) Mathematics syllabus and the other uses formulas for motion under constant acceleration taught in Physics.  Whichever method is used, remember to convert from km/h to m/s.  The target velocity is  ⁷⁰⁰/₉ m/s,  and the time to achieve this is  ⁷⁰⁰/₉ s  starting from rest,  since the acceleration is  1 m/s².

Solution 1 [“Elementary” Mathematics, speed-time graph]

     The speed-time graph is very useful because it is able show the acceleration (as the gradient or slope of a straight line) and at the same time the area under the graph gives the numerical value of distance travelled.  If the acceleration is constant, we usually we get a trapezium.  But since the aeroplane starts from rest, we get a triangle (see diagram below).  All we need to do is to calculate the area under the graph and get the answer.
     It is interesting to observe that if we used the average speed  ⁷⁰⁰/₁₈ m/s,  we would also get the answer because the area under the graph (yellowish green rectangle) is the same as the area of the triangle.  This trick works for constant acceleration, but it may not work in other situations.

Solution 2 [Physics, constant acceleration]

     We use the important formulas  v = u + at  and  s = ut + ½at².  In our example,  u = 0  because the initial velocity is zero (the airplane starts from rest).  This makes our calculations very easy.  If we compare the two methods, you find that the calculations are very similar, and we get the same answer.  Remember that speed = |velocity|  the magnitude of velocity.  In this relatively easy problem, the velocity means the same as speed because we are going in a straight line and in one direction only.  In other situations, this may not be so.

Heuristics Used

H02. Use a diagram / model

H05. Work backwards

H09. Restate the problem in another way

H13* Use Equation / write a Mathematical Sentence

Suitable Levels
* GCE ‘O’ Level “Elementary” Mathematics
* GCE ‘O’ Level Physics
* other syllabuses that acceleration, speed and distance
* precocious kids who always want to learn more

[S1_20150408DST] Funky Bus between Towns

Question

Introduction

     This is quite a convoluted word problem for a Secondary 1 (roughly equivalent of grade 7) pupil in Singapore.  There are a lot of data given and lots of unknowns.  How shall we cope with this?  One good way I recommend is to use a table to organise the information.  [H02]   Many primary school teachers teach the Triangle Mnemonic for the relationship between Distance, Speed and Time.  This continues to be useful in secondary school.

To get Distance, cover “Distance” with your finger, and you get Speed ´ Time.

To get Speed, cover “Speed” with your finger, and you get Distance ¸ Time.

To get Time, cover “Time” with your finger, and you get Distance ¸ Speed.

Solution

Remember: Tables are very useful to help to organise information before you try to solve the problem.  They also help you to formulate the algebraic equations correctly and quickly.

Heuristics Used

H02. Use a diagram / model

H10. Simplify the problem

H11. Solve part of the problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels
* Secondary 1 Mathematics
* precocious primary school pupils
* anyone, young or old, who enjoys an algebra challenge

[Maths History] #Newton's #proof that #pi is #transcendental ?

Article (from Quora, re: Alejandro Jenkins)
How do you prove that a number is a transcendental number?

Summary
Although the concept of "transcendental number" was not defined during Newton's time, Newton's Principia contained the germ of idea that seems to lead to a relatively easy proof of the idea that p  is transcendental.  Arnol'd described how Newton showed that the trajectory of planets cannot be expressed in terms of any polynomial function of time.