1) In a recent winter Olympics, 59 gold medals were awarded to Canada, Austria, Norway, the Netherlands, Japan, Italy, Russia, and the United States. A journalist, in summarising the games for his editor, noted the following facts:

• Norway won five times as many gold medals as did Russia.

• Russia won more gold medals than did Italy.

• If Japan had won four more gold medals than in fact they did, then they would have won the same number of gold medals as Canada.

• The Netherlands won eight more gold medals than one-third the number of gold medals won by Austria.

• If Canada had won one more than three times the number of gold medals than they did, then they would have won thirteen more gold medals than the United States.

Your task is to work out the number of gold medals won by each of the eight countries.

Q:a) An important part of problem solving is to first see what restrictions there are so that you do not waste time trying to construct a solution that can never work. Examine the facts above VERY CAREFULLY and list as many restrictions on the possible medal count for each country as you can, explaining your reasoning in each case. To start you off, it is worth noting that each country has won at least ONE medal.

b) Give any solutions you have for the above scenario, with a clear explanation of how you reached that solution.

c) Explain clearly what difference it would make to your answer to (b), if the second criterion above was changed to “Russia won exactly ONE more medal than did Italy”.

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2) “LIDAR (Light Detection and Ranging; or Laser Imaging Detection and Ranging) is a technology that determines distance to an object or surface using laser pulses. Like the similar radar technology, which uses radio waves instead of light, the range to an object is determined by measuring the time delay between transmission of a pulse and detection of the reflected signal.” (LIDAR - Wikipedia, the free encyclopedia).

Your job is to write the software for a new laser ranging device and to write the user manual. The device uses “eye-safe” laser light of wavelength 750 nanometres, sending pulses 1200 times every second. The speed of light is (in round numbers) 300 000 km per second. The following questions are about some of the situations you will encounter in carrying out these tasks. (You will need to research any new terms that you do not understand).

Q: a) The device relies upon a very sophisticated timing system which you must calibrate as part of the software. How long would it take a pulse of laser light to travel to a piece of reflective tape 5 m away and back again?

b) The laser goes through a PULSE – NO PULSE cycle 1200 times a second. The duration of the no-pulse is the same as the duration of the pulse. You need to calculate these three things:

(i) The time a single pulse lasts.

(ii) The distance the pulse travels in that time = the length of the pulse.

(iii) The number of complete waves in the pulse.

c) The lasers are also going to be used as part of a burglar alarm system in huge museum galleries around the world for large archaeological displays.

The end walls, such as CDEF in the diagram above, are 4m by 4m in size. Laser beams will stretch across from diagonally opposite corners (an example is the line BF shown). This particular laser’s distance measurements become less reliable beyond 30m. Construct a mathematical model (using Excel or otherwise) to show how the length of a diagonal increases as the length of the hall increases (eg BF compared with GF) and show your model as a graph. Use this to find the longest the museum hall could be before the laser would exceed its range.

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3) Orlando Spoon has long been a fanatical developer of automated devices to enter into “Robot War” race games. (He did take one year off to play the husband of Lego Lass in the Internet movie Lord of the Pings). Two of his latest runners are shown above. They are named H (for “HARE”) and T (for “Tortoise”). H has a top speed of 1.6 ms–1 which makes him a formidable competitor. Unfortunately, a fault in one of its logic chips causes a software error that limits T’s top speed to 0.2 ms–1 .

Q: a) Orlando argues that T can win any race if he is given a bit of a head start. He argues this way. “I start the race with T 80m ahead of H. When H has travelled this 80m, T will have gone 10m and still be ahead. When H has travelled this 10m, T will have gone 1¼ metres and is still ahead. When H has travelled this 1¼ m, T has travelled …” and so it goes on. Orlando thinks he has proved that H will never actually catch up with T. Explain how Orlando’s argument works and take it on two more steps. Then, try to find the flaw in the argument, giving clear reasoning

b) Orlando decides to test his idea. He places T 80m ahead of H and starts them off simultaneously with a radio signal. He accidentally places T so that he is facing the wrong way. H and T hurtle towards each other at their top speeds. How far has T travelled when they crash into each other? Be sure to explain your reasoning.

c) Orlando sets up the test again, T 80m in front of H and facing the same way. Once again, a radio signal starts them off. Orlando sees H catch up with T and pass him. Show the motion of H and T on the same graph (assuming they go at their top speeds throughout), showing how far each robot travels in equal intervals of time. Use this graph to discover how far T has travelled when H overtakes and to discover how much time has elapsed. Explain the reasoning.