Sunday, November 21, 2010

Sunday November 21, 2010

  • A thick metal pipe,  2mts long, has inner radius of 12 cms and outer radius of 15cms, is made of steel. How much volume of steel is used to make this pipe?  What do you think is the weight of this pipe? Note that weight of 1 cubic meter of steel is 8000 Kg.  What if the pipe is made of Bamboo - 1 cubic meter of Bamboo is approximately 350 Kg.
  • The new Chapman Mills park is rectangular, where the sides are in the ratio of 4:3. The total area of this park is approximately 3500 sq mts. What is the cost of installing a metal fence around this park, where The Ottawa Fencers charges approximately $45 per linear meter.
  • Ms. Quick finishes a job in 8 days, whereas Mr. Slow finishes the same job in 12 days. How long will it take if both of them will work together (assuming that the job can be partitioned nicely (like pulling weeds in the garden)).
  • Dr. Light needed to climb his wall to install the newly bought Christmas lights. His ladder is 8mts long, and the wall is approximately  7mts long. How far is the base of the ladder from the wall? Do you think its at a safe distance and will he be stable on the ladder?
  • Suppose we toss two unbiased coins 100 times and observe that the number of times we get 2 heads is 25 and number of times we get no heads is 18. How many times we got at least 1 tails? We can try the same problem with 3 coins - lets say #times 3 heads= 22, #times 2 heads=32, #times 1 heads = 24, #times 0 heads = 22.   How many times we get at least one tails, at least one tails and one head.
  • A disc needs to be cut out from a square sheet. The length of the sheet is 2mts. Whats the area of the largest possible disc? Whats the area of the left out piece of the sheet? Try the same problem for a cube and a sphere. Is the volume of the left out piece larger than that of the sphere?
  • The angles of a triangle are in the ratio of 1:2:3. What are the angles?  Is it a right angle triangle? What if it is 3:4:5?
  • Mr. Too Quick puts all his favorite NHL-team socks in the drawer, and in the morning rush (in the dark in long Northern Winter)  - what are chances that he will draw a pair which is of the same team. He has 2 pairs of SENS, 2 pairs of PENS, 2 pairs of  DUCKS and 3 pairs of HAWKS. What are the chances that he will draw a pair which does not belong to the same team?  First try this for two teams, and lets say one pair from each team, and then start to make it more complex.

Sunday, November 14, 2010

November 14, 2010

The first one is from Mathematics Olympiad Problems (1996 Vologda). The probability ones are inspired from
  • There are 3 boxes, each consisting of two balls. One of the boxes contains exactly two white balls, one consists of exactly two red balls, and the third one consists of exactly one white and one red ball. But while labeling the boxes, their labels got mixed up so that none of the boxes has the right label. Is it possible to identify  which  box contains what by just drawing exactly one ball from each box? Can it be done by drawing just two balls or one ball?
  • Since one of the kids in our team is a president of his class, he needs to come up with creative ways to do fund raising. What could be a better way then doing some kind of lottery!  Mr. President decided to setup the lottery as follows. In a bag, he places 4 identical hockey pucks in a bag, and numbers them from 1 to 4. Each kid (there are 24 of them who wants to make fortune!) walks up to him, hands him $1, and makes a guess from 1 to 4, and draws the puck. If the guess works, that kid gets $2, otherwise looses his money for the fundraiser. Do you think whether Mr. President will raise any funds? (BTW - kids can't gamble!)
  • In the above setup, kids are asked to guess two numbers, instead of one. Then they draw two pucks and if the numbers on the puck matches with their guess, they win. First they draw a puck - record what is the number, places this puck back in the bag and then draw the second one (i.e., with replacement). For example if I guessed 12, then either I draw (1,2) or (2,1), I win. Do you think whether President will raise more money by this scheme. What happens when it is without replacement?
  • What is the probability that two kids in the same family have a birthday on the same day?  There are approximately 30,000 families in City of Waterloo with two kids. How many of them will likely have kids having birthdays on the same  day.  There are approximately 14,000 families with three kids. What is the probability that all three of them will have their birthday on the same day.
  • You are given a right angled triangle. You need to draw a line parallel to its base, so that the triangle is split in two parts of equal area. Where should one draw the line?
  • There is a study done in Ontario schools, in terms of whats the average number of hours/per week the students in Grade 9 spend on their homework. Here is the data: 40 % don't bother, 10% spend 1-5 hours, 10% spend 6-10 hours, 20% spend  11-15 hours,  18% spend 16-20 hours and the rest spend more than 21 hours. Whats the average time a grade 9er spend on their homework?
  • Consider the pattern:  The first term when n=0 is 2, when n=1 it is 8, when n=2 it is 14,  when n=3 it is 20, ... Once you figure out this pattern, you will realize that it is a linear pattern. You can graph it and see that it is a straight line.  For a pattern which grows linearly, show that it is sufficient to have two consecutive values to find the pattern rule.  Consider a pattern which follows a quadratic growth, for example the pattern when n=0 it is 1, when n=1 it is 2, when n=2 it is  5, when n=3 it is 10, ...17, .. 26,... Show that it is sufficient have three consecutive values to determine the pattern rule.  How does the graph of these patterns look like?

Sunday, November 7, 2010

Sunday, November 7, 2010

  • Consider three unit disks (i.e., of radius 1) touching each other. I am interested in finding the area of the triangular cavity between the three disks. Draw the triangle joining the center of the disks, and find the area of the part of the triangle, which is outside of the disks.
  • Consider a cube, which is painted red, and has side length of 4cms. This cube is divided into unit cubes (i.e., each of side 1cm). How many small cubes will have (a) exactly three sides painted red (b) exactly two sides painted red (c) exactly one side painted red (d) no sides painted red.  Repeat the same problem with the starting cube of side length 1cm, 2cm, 3cm, .. Whats the pattern?
  • Ad's friend decided to hangout on Friday evening in the Ottawa Expo. They planned carefully and decided that they needed to spend $96 for all the fun activities - and decided to contribute equally. But on the most exciting day, some of his friends (being very unreliable) didn't show up.! Now the remaining ones had to contribute $4 more than what they initially planned! How many of them were eventually at the expo?
  • Mr Every T. Wrong is expert in getting everything wrong! He was asked to multiply a number by 4 and then add  eight - and he landed up dividing by 4  and then subtracting 8. He got 1 as his answer! What should be the correct answer.
  • The overhead water tank on the Moodie drive supplies water to whole of Barrhaven.  It requires 4 hours to fill it completely and  6 hours to empty it. Usually the water tank is getting simultaneously filled as well as emptied. How long will it take to fill it? Suppose it takes x hours to fill and y hours to empty, what should be the relationship between x and y. This is a typical producer-consumer problem! Assuming that x is less than y, then how many hours it will take to fill the tank completely in terms of x and y?
  • Ad is taller than Rd by 25%. Then, by what percentage Rd is shorter than Ad?
  • The height of parliament hill in Ottawa is 92 mts and the height of the National Art Gallery  is 60 mts. They are approximately 200 mts apart. Assuming that their base is at the same level,  what is the distance between their tops? 
  • You have a bag consisting of 10 red and 10 blue marbles. What is the smallest number of marbles you need to draw to be sure that you have exactly three of the same color? What happens when you have 10 red and 15 blue? What happens when you have three colors - say consisting of 10 red, 10 blue and 10 green - and you want to draw three of the same color?
  • Whats the right length of laces that you need for your ice skates?  You need about 20  cms  to be left after tightening your skates to tie them up. There are 10 eyelets on each side,  3cms apart, and the distance between  two consecutive eyelets on the same side is about 1 cms.