WHEN THE MEASUREMENT GOES OUT OF HAND ....

A severe earthquake is many many times stronger than a shake of the table. How to measure both of them in a scale from 1 to 10? The sound of a bomb blast is many times highly intense than a whisper. How to measure them in a scale of 1 to 100 ?
Here logarithms comes to our help.

we know,
10^2 = 100
log 100 = 2.0000
(base 10)
2.0000 is the logarithm of 100 to the base of 10
similarly,
log 10000000000 = 10.0000
(base 10)

Hence using the idea of logarithm, we can represent any number lying between 100 and 10000 crores in the scale of 2 to 10.
example:
log 102456 = 5.010537
(base 10)
Long range of numbers can be compressed using logarithms.

In the case of earthquake, the formula is,
R = log (A/A0)
A0- the intensity of smallest detectable shock wave by seismograph.
A - the intensity of earthquake wave
If we get R as 3, the earthquake is 1000 times intense than the smallest disturbance. Since log 1000 = 3.0000 This scale is popularly known as Richter scale.

Formula for measuring sound intensity
d = log (P/P0)
P0 - weakest sound human can hear.
P - real intensity of sound.
Here the unit is decibel. D= 50 decibel means the sound is 10000 times stronger than a murmur.

Logarithms is originally invented to make numerical calculations easy. But the advent of electronic calculators and computers made it obsolete. But the logarithmic function is still useful as above.

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LISSAJOUS FIGURES

Definition: "When a particle is subjected to two sine wave motion or two oscillatory motion at right angles, the particle describes lissajous figures". We know sine wave motion and circular motion is basically same. Hence we draw two circles A and B perpendicular to each other. The circle B rotates twice faster than circle A. That is, frequency of circle B is two times than that of A. A particle at the intersection of two circles is subjected to two sine wave motion A and B at 90 degree simultaneously. The particle will describe figures depending on the frequency and phase of A and B . In our case, the ratio of frequency is 1:2 and the two waves are in phase. To draw lissajous figures : A moving point in both the circles are chosen. Here we should remember; during the time taken by the circle A to complete one rotation, circle B completes two. Hence the points are marked on the circles according to their speed. Then straight lines

THE PARABOLA

A jet of water shooting from a hose pipe will follow a parabolic path. What is the so special about parabola. Y= x^2 Draw a graph for the above equation. It will result in a parabola. This parabola is also called unit parabola. Any equation involving square will yield a parabola. Example: Y = 2x^2 +3x+3 (also called quadratic equation) X= 2 and -2, both satisfies the equation 4 = X^2. Parabolic equations always have two solutions. Any motion taking place freely under gravity follows parabolic path. Examples: An object dropped from a moving train, A bomb dropped from flying plane, A ball kicked upwards. If a beam of light rays fall on the parabolic shaped mirror, they will be reflected and brought to focus on a point. This fact is made use of in Dish Antenna, Telescope mirrors, etc. Inverted parabola shape is used in the construction of buildings and bridges. Because the shape is able to bear more weight. A plane

CASINO'S GAME

Let us find out how the casino survives with mathematics. Say, your friend invite you for a game of dice. You must bet (wager) 2 dollars. If you roll 'six' you will get back 8 dollars. The game will go on for 30 rounds. All sounds good. The probability of rolling 'six' is 1/6. Since the game will be played for 30 times, the 'expected win' is 30*1/6 = 5. That is, you are expected to win 5 rounds out of 30. Hence your gain will be 5 * 8 =40 dollars. ok. This also implies that you will loose 25 rounds. Hence your loss will be 25*2 =50 dollars. Your net gain will be gain-less = 40-50 = -10 dollars. For 30 rounds, the loss is -10 dollars, Hence, for one round =-10/30 = -1/3 dollars. There will be a loss of -1/3 or 0.33 dollars per round. It is not a fair game. Let us make a simple formula to calculate 'Pay out per round\. The probability for a win = p The pay-out in case of win = V No. of rounds = n The expect

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