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GOMBOC: THE SHAPE THAT CANNOT EXIST

        VISIT:  STATISTICS LAB ONLINE   for interactive and animated experiments.    Consider an egg. You can balance it on one of its ends. It won't last long. The slightest disturbance will topple it. The two tips of the egg are known as unstable equilibrium points. "Equilibrium" because you can balance the egg on them and 'unstable' because the egg will fall over at the slightest nudge. The points around the egg's body on which it balances robustly are called 'stable equilibrium' points.      Now consider a flat square (a cardboard piece). It has four 'stable equilibrium' points at the center of the sides. You can balance that square on these points. It also has four unstable equilibrium points at its corners. Similarly, a triangle has three stable and unstable equilibrium points. All ellipse has two stable equilibrium points in the centers of its long side and two unstable ones at the ends.     Can you find a two or th...

HOW TO 'ESCAPE' FROM A CLOSED ROOM?

      VISIT: STATISTICS LAB ONLINE   for interactive and animated experiments.     Put a coin in a plastic tray. Consider both as a two-dimensional object. The coin can move within the tray but cannot escape from it. Suppose you lift the coin from the tray, it escapes from the boundaries of the tray. That is, you have moved the coin into the third dimension, that is why it escaped.      Now put a marble in a box. Both are 3D objects. You cannot move the marble outside the box without damaging the walls of the box. But, if you move the marble into the 4th dimension, the container walls cannot stop its movement(as in the coin and tray- case) and the marble will easily escape from the containment of the box. Similarly, one can escape from the jail, if be could move into the fourth dimension.      Einstein said, that besides three dimensions(length, breadth, depth) the time can be considered as the fourth dimension. We are abl...

SOLVING A REAL LIFE PROBLEM WITH A PICTURE

     John lives in a warm country. He has just 30 square meters of land. He can plant cocoa and pineapples. How much of pineapple and cocoa he has to plant in his plot to get maximum profit? This is a classical problem. It can be solved with graphs.     Let x stand for the amount of cocoa and y for the amount of pineapple. When john harvests the crops, he will get a unit return of 3 dollars on the cocoa and of 2 dollars on the pineapples. His total return p will be  p = 3x+2y       ----(1)     The profit p, we want to maximize. Planting comes with a cost. The unit cost of cocoa seed is 2 dollars and of pineapple, the seed is 1 dollar. Also, 4 dollars is required to cultivate both the crops in his land.     Hence, the total cost c of growing the two crops is       c = 2x+y+4   -----(2)      John has only 50 dollars in the bank. That is the first constraint. We write it mathe...

WHY THE SCHOOL TOPPER DOES NOT BECOME COLLEGE TOPPER?

     ADVT:  SELL ON AMAZON      Think of Usain bolt. He created a world record in 100m dash. Could he repeat the record performance the next day after winning gold? Probably, he could not. Because many factors come into play on that particular day like his mood, the condition of the track, the passion of the crowd, etc. Things will not fall into paces again.      After an extreme performance, all of us may fall back to our average or around the average. That is why, a student who excelled in school final year, may not make it at the college level.     We have our own average values of BP, sugar, temperature, speed of writing, speed of running, response time, etc. When running, the BP value may increase, after a while, it will fall back to 'our average'. This is called"regression to mean" mathematically.      The above is the probability curve drawn between performance and its quality. Mostly, our performance clus...

ENERGY - IN A NEW ANGLE

      ADVT:  SELL ON AMAZON       A car is running. It can hit and damage an object. It can do some work because it posses energy. A body gets energy by virtue of its motion or its dynamics. We know, it is called kinetic energy. The formula is 1/2 mv^2, m is the mass of the body, or we can say m is the inherent property of the body. v is the velocity of the body. or we can say v represents the dynamics of the body. Almost all the kinetic energy formula in physics follows this pattern.      Consider an oscillating spring. It is KE is 1/2kx^2. Here k is constant (property) of the spring. It depends on the material, thickness, and length of the spring. x is the maximum displacement of the spring from its rest position.  We say x represents the main dynamics of the spring.     Consider a rotating body. Its K.E is 1/2IW^2. I is the moment of inertia or property of the rotating body, w its angular velocity (dynamics).   ...

THE CATCHMENT AREA FOR YOUR BUSINESS

      In a medical emergency, you always want to rush to the nearest hospital. A city has a number of hospitals. Suppose we have a map showing each hospital catchment area (for anyone in that region, that hospital is closer than any other) that will be highly helpful. How do you make that map?     It can be easily done. But doing it manually is difficult. With a computer algorithm, we can easily create a map.      Start with two hospitals at points A and B on the city map. Draw a line that connects them. Find the midpoint of that line and then draw a line that passes through that midpoint and is perpendicular to the line from A to B. That line divides the city into two regions. One region having A contains all the points closer to A than to B. The other contains all the points closer to B than to A.   ADVT:  TODAY'S DEALS IN AMAZON Now, look at the third hospital, at point C. Repeat the above process to the hospitals at A and C, you get...

A FORMULA TO FIND A SERIAL KILLER

      Is it possible to create a formula to locate a serial killer? Yes, one man has done it. Mathematics looks for a pattern. Serial killers leave some pattern. From that pattern, we can make a formula.      Serial killers strike some distance away from their residence, but not too far away.   ADVT:  TODAY'S DEALS IN AMAZON     We select a large area where the killer is supposed to be hiding and map it. We superimpose the map on a square grid. Each square can be located using x, y coordinates as shown in the figure. We select a particular square(shaded one) and we will find the probability that "the killer will be residing there" using a formula. The formula is              The formula seems more threatening than a serial killer. Anyway, we will go step by step and understand the formula. Pij- - represents the probability finding the killer at the location i, j (x,y coordinates). The answer range...