Q.47.

[Manish Salian]

Hi,

I have extended the deadline for Q.49. So, there is no change from yesterday's scores...

Here is today's question:

Q.47.

A group of 12 sailors are travelling in a ocean liner, when they discover big hole on one of the walls of the liner. They immediately start the rescue operation and send a SOS. Within minutes a rescue ship arrives at the spot and they start moving the sailors into the new ship - one at a time. They observe that, the water level is just 25 meters below the hole when they just begin to shift the first sailor.

It rises by 10 meters in the first second and every second thereafter the water level rises by 60% of the height risen in the previous second.

Also, it takes 10 seconds to rescue the first sailor and for every subsequent sailor it takes 80% of the time taken to rescue the previous sailor.

The question is, will they be able to rescue all the sailors? If no, how many sailors will be saved? (Note that only those many sailors are assumed to be saved whose rescue operation is fully completed till the water level rises to the level of the porthole).

Yes, for a change a math funda oriented question!! Start your calculations now!

Deadline: Wenesday midnight

Regards,

Manish sir

[PiyushTantia]

Hello Sir,
The answer for the above would be that they will be able to rescue all the sailors.

Explanation:

1st sec the water level rises by 10 m
2nd sec by 6 m
3rd sec by 3.6 m and so on

so let us take 10 as x and solve it in an equation form

x + 0.6x + 0.6(0.6x) + 0.6{0.6(0.6x)}........

so basically what we get is

x + 0.6x + 0.6²x + 0.6³x + .................

x(1 + 0.6 + 0.6² + 0.6³ + ...........

this is equal to

10/1-0.6 (an engineering student told me abt this)

which is equal th 25

that means infinite number of seconds are required to sink the ship entirely by 25 metres

hence 12 sailors who take 46.56402616 seconds to get rescued can be rescued.....easily

[Manish Salian]

Hi Piyush,

Although you have given the correct answer, you are eligible for only 2 points as I am not satisfied with your explanation.

Please provide the correct explanation and win 3 more points!!

In case someone gives the correct explanation, he wins 5 points...

Remember, the deadline is today evening!

Regards,

Manish sir

[AnilkumarBDaga]

Hello Sir,

All the sailors can indeed be saved. The explanation is given below:

The time taken by each sailor to be saved right from the 1st to the 12th sailor is:

10, 8, 6.4, 5.12, 4.096, 3.2768, 2.62144, 2.097152, 1.6777216, 1.34217728, 1.07374182, 0.85899346.

Hence the total time taken is just a shade less than 47 seconds.

BTW the above series is a GP with the first term as 10 and common ratio as 0.8. The number of terms being the number of sailors i.e. 12. (I hope this is what you were looking out for Sir).

Similarly the second series of the rise in water level for 47 seconds is also a GP. (47 because this is the time which will be taken to rescue all the sailors from the ship).

The variables for this series of GP would be n=47 (seconds), a (first term) = 10 and r (common ratio) = 0.8.

This gives us the sum of the series to be a little less than 25. (which means the ship will sink a little less than 25 meters till the time all the 12 sailors are rescued - and hence would not sink).

If we want to use the conventional formulae for calculating the second series we would require the 48th term of the series (and hence for ease of calculation we could assume it to be an infinite series). The formulae for this has been used by Puish above. That formulae gives us a value of 25. But we know that 47th term would be something greater than zero and hence reducing the numerator (while denominator remains constant) and the final answer would also be a little less than 25 (which again means that the ship won't sink).

BTW the following link gives us the formulaes relating to GP and their derivation as well - http://en.wikipedia.org/wiki/Geometric_series

Thanks & Regards,
Anil Daga

[RobinRaju]

all the sailors will be rescued as the boat will rise with the water level rising
hope that its right sir :rolleyes:
thanks,
ROBIN

[PiyushTantia]

Sir,
even Anikkumar got what i wanted to explain.
neways all i wanted to say is that the sailors take less than 47 seconds(the exact time was given in the previous explantion) and the split up of time taken by sailors is given by anil. let me give you the split up of the level of water rising.

Seconds Level of increase
1 10
2 6
3 3.6
4 2.16
5 1.296
6 0.776
7 0.46656
8 0.279936
9 0.1679616
10 0.10077696
11 0.060466176
12 0.0362
13 0.02172
14 0.0130
15 0.0078192
16 0.00469152
17 0.002814912
18 0.0016889472
19 0.00101336832
20 0.000608020992


so this adds up to 24.90............(something) but we can see that the number of decimals are constantly increasing and the number of zero's will keep increasing after the decimal, its difficult to even reach 24.999 after adding each of those. Every next number tht will be added in more decimal numbers and its going to take a lot(means really really lot) of time in seconds(almost infinity) to add upto 25metres. the formula was explained earlier. Thus it can be very very safely assumed that atleast till 47 seconds the ship will not sink.

Huh!

Is that okay
If nt temme i'll come and show you the complete working till 47 seconds.

nw can i have those 3 points and what sir, before the deadline is met you're declaring the answer. What if other people had given some other answer.

kya sir, isme bhi politics!

[Manish Salian]

Yes Robin! That is the answer I was looking for!

The boat will rise along with the water, hence there is no question of the water level reaching the hole!!

Piyush and Anil, you did not require all those calculations!!!

At times, common sense work better than Maths...

Robing, you earn yourself 5 points, Piyush and Anil you get 2 points each.

Regards,

Manish sir