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#41901 Re: Jai Ganesh's Puzzles » Problems and Solutions » 2005-08-04 16:29:22
Great work, Mathsy!
I don't know how you solved the problem, but this is how I would do it.
Let 'l' denote the light that strikes the surface of water at depth 'h' then
dl/dh = -kl
dl/l = -kdh
Integrating,
log l = -kh + c
Let log l = lo when h=0, so that
log (l/lo) = -kh
l/lo = (100-40)/100 = 3/5 when h=10
therefore, log 3/5 = -10k --------------------------(I)
We have to find h when
l/lo = 1/300,000
log (1/300,000) = -hk -----------------------------(II)
Dividing II by I,
-log 300,000/ log 0.6 = h/10,
h=247 metres approximately.
#41902 Re: This is Cool » Digit-perfect Numbers!! » 2005-08-04 16:17:43
Superlative work, Mathsy 
How did you do it?
#41903 Re: Jai Ganesh's Puzzles » Problems and Solutions » 2005-08-03 16:05:30
Problem # n+1
We know water absorbs light by the fact that it is dark in the ocean depths. If 10 metres of deep water absorbs 40% of the light which strikes the surface, at what depth would light striking the surface be reduced to 1/300,000 th? Assume that the rate of absorption is proportional to the light.
#41904 Re: Jai Ganesh's Puzzles » Problems and Solutions » 2005-08-02 21:00:34
Mathsy is right This is the complete solution.
Assuming my normal speed is 's' and the normal time taken is 't',
Distance/Time = Speed (or) Distance/Speed = Time
We get 840/s = t ---------------------(1)
The second equation would be 840/(s+5) = t-3
840 = (t-3)(s+5)
840 = st+5t-3s-15 -------------(2)
Subtracting (1) from (2) 840 = st we get
0 = 5t-3s-15
Putting t = 840/s (from (1), 0 = 5(840/s) -3s - 15
0 = 4200/s - 3s - 15
0 = 4200 - 3s² -15s
3s² +15s - 4200 = 0
Dividing by 3 s² + 5s - 1400 = 0
s = [-5 ± √(25+5600)]/2
s = [-5 ± √(5625)]/2
s = (-5 ± 75 )/2
Taking the positive value, s = 35
Therefore, my normal speed is 35 kilometres/hour.
#41905 Re: This is Cool » Easy problem » 2005-08-02 19:11:24
Later, I found on the same website that I originally found the problem a proof showing there was a single unique solution.
Can you post the proof? I got as far as Mathsy did, that two of the numbers should be odd and one even. Maybe, the proof has got something to do with 2 being the only even prime
#41906 Re: Jai Ganesh's Puzzles » Problems and Solutions » 2005-08-02 16:10:54
Problem #n
The distance between two cities is 840 kilometres.
If I start driving from one city to the other at a speed
5 kilometres/hour more than my normal speed,
I save 3 hours. What is my normal speed?
#41907 Re: Jokes » Technology Jokes » 2005-08-01 16:26:48
Mathsisfun has proved that he is a Computer Scientist 
#41908 Re: Help Me ! » Has anyone seen this formula before? » 2005-08-01 16:21:31
How did you get to the c² - 2bc (cos θ) + b² (cos θ)² ?
You can expand the terms as Mathsisfun said.
Or, you may remember the following formulae:-
1. (a+b)² = a² + 2ab + b²
2. (a-b)² = a² - 2ab + b²
3. (a+b)(a-b) = a² - b²
4. (a+b)³ = a³ + 3a²b+3ab²+ b³
5. (a-b)³ = a³ -3a²b +3ab² - b³
6. a³ + b³ = (a+b)(a² -ab +b²)
7. a³ - b³ = (a-b)(a² +ab + b²)
#41909 Re: Introductions » hi » 2005-08-01 16:12:03
I had read somewhere that animals (the most intelligent ones) cannot count beyond 7 or 8! They are good at counting up to 4, thereafter, there is a progressive decline in their ability to count, and beyond 8, they are unable to count! 
#41910 Re: Puzzles and Games » Trains » 2005-07-31 23:03:23
There's no inactive train at all! There is only one train!
The train passes through the tunnel, heading for the park, at 8 AM.
The same train passes through the tunnel, going away from the park, at 8 PM.
#41911 Re: Help Me ! » Has anyone seen this formula before? » 2005-07-31 22:59:05
Oh ok. But what formula would I use if i want to find the angle oposite to side 'b' if i'm only given the lengths of sides 'a' and 'b' and the angle between them?
With the help of the formula
c² = a² + b² - 2ab Cosθ,
you can find the value of c.
Now, use the formula
b² = a² + c² - 2ac Cosθ.
You know the value of a,b, and c.
You would get
Cosθ = x (some value)
The angle opposite to side 'b' would then be
θ = Cos-¹x.
Is that clear?
#41912 Re: This is Cool » Which of the two is greater? » 2005-07-31 22:21:46
I've got as far as the yth root of y = the xth root of x, but now I'm stuck.
Interestingly, if yth root of y = xth root of x,
it does not automatically follow that x=y
For example, if x=4 and y=2,
then this is true!
#41913 Re: Help Me ! » Has anyone seen this formula before? » 2005-07-31 22:15:20
Zmurf,
This formula is easier remembered as
a² = b² + c² - 2bc cosθ
On simplification of the equation given by you, this is what is obtained.
a,b, and c are the three sides of a triangle.
If the triangle is rightangled and θ is 90 degrees or pi/2 radians,
Cos θ = 0,
which gives us
a² = b² + c²,
the Pythogoras Theorem.
#41914 Re: Help Me ! » what u think of this problem » 2005-07-31 21:03:13
Corrected the post; thanks, Mathsyperson.
#41915 Re: Guestbook » Everyone has to like him » 2005-07-31 20:57:46
You are right, Rora! Mathsisfun is remarkable
#41916 Re: Dark Discussions at Cafe Infinity » Be aware of my absence » 2005-07-31 20:08:39
Welcome back, Rora
#41917 Re: This is Cool » will tehre ever be a single particle that makes EVERYTHING » 2005-07-31 18:58:40
But before that read 'A Brief Histroy of Time' by Stephen Hawking
#41918 Re: This is Cool » the philosophy and science behind teleportation » 2005-07-31 18:55:51
Mathematics lovers discusing Futurology and Genetics (and Cybernetics?)
#41919 Re: Jai Ganesh's Puzzles » Problems and Solutions » 2005-07-31 18:54:04
As always, you are corect, Mathsy!
I shall show how I did it.
Assume both candles are 'h' inches in height.
The height of the first candle reduces by h/4 every hour and that of the second reduces by h/3.
Lets assume in time 't' hours, the first is twice the height of the second.
Therefore,
h - t(h/4) = 2 [h - t(h/3)]
h - ht/4 = 2h - 2ht/3
h = 2ht/3 - ht/4
h = 5ht/12
Cancelling h on both sides,
1 = 5t/12
or t = 12/5
The unit we had taken was hours,
therefore, in 12/5 hours, that is 2 2/5 hours, one will be double the height of the other.
Simplified, it is 2 hours and 24 minutes.
#41920 Re: Help Me ! » what u think of this problem » 2005-07-31 18:36:24
Let's plug in some numbers just to get this started:
x=1: 3x^2 + x = 4y^2 +y becomes 5 = 4y^2 +y
When we put x=1,
3x² + x = 3(1² ) + 1 = 4
Therefore,
4y² + y - 4 = 0
y = [-1 ± √ (1 + 64)]/8 = [-1 ± √ 65 ]/8
which is an irational number.
We see that here too x-y is not the square of a natural number
#41921 Re: Help Me ! » what u think of this problem » 2005-07-31 18:21:00
x and y are two natural numbers such that 3x^2 + x = 4y^2 +y
Put x=2,
we get 4y² + y - 14 = 0, Solving,
we get y = [-1 ± √(1 + 224)] / 8
y = -2 or 1.75
Neither of them are Natural Numbers!
x - y is NOT always a square of a natural number
#41922 Re: Jai Ganesh's Puzzles » Problems and Solutions » 2005-07-29 19:54:39
Two candles A and B, of equal height but different circumferences, burn for 4 hours and 3 hours respectively. If the two are lit at the same time, after what time would one candle be half the height of the other?
#41923 Re: Help Me ! » Hi » 2005-07-29 18:22:15
Nora can paint 1/6 of a house in 1 day.
Therfore, she takes 6 days to paint the house.
Meera can paint the whole house 2 days faster.
So, she can paint the house in 4 days.
In one day, Meera can paint 1/4 of the house.
If both of them paint the house together, in one day,
they can paint (1/6+1/4) of th house. That is,
10/24 or 5/12. Therefore, the whole house can be painted
by them in 12/5 days, that is 2 2/5 days.
#41924 Re: This is Cool » the philosophy and science behind teleportation » 2005-07-28 18:27:54
We are discussing this 2.500 years ahead of mankind
Maybe in 4,500 AD, brain and the body could be digitized,
maybe then the DNA pattern would be
010100101011100101011001011100011010100010110001.........................
and not
AGCTGGCTAACGTTACTGTTACACGTACGGAATCAGTTAAGCTAAGCT..........
#41925 Re: Dark Discussions at Cafe Infinity » do fish have a personality » 2005-07-28 16:51:29
i don't think fish have real process of their mind, it is proven that they don't feel pain.
It is wrong to assume fish don't feel pain. Any member of the animal kingdom that reacts to touch does feel pain (I think )