reinout-g

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 #26
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+11
Here are the new probability puzzles!
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Easier one:
A special machine contains 100 red, 100 green and 100 blue marbles.
The machine performs the following four operations.

First, the machine randomly picks 50 marbles from the entire batch.

Secondly the machine divides the marbles into one group of 20 and one group of 30

Furthermore, the machine picks and shuffles 5 marbles of the group of 20 and 10 marbles of the group of 30.

Finally, the machine randomly chooses one of the final 15 marbles it picked and shows it to you.

Assuming you cannot see anything which happens inside the machine.

What are the odds the machine will show you a red marble?

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Hard one :

Steve has two routes (A and B) to travel to his girlfriend which he is indifferent about (see my amazing illustrational skills )

knipsel11.png

Steve has decided to play a game which makes him decide which route to pick.
He always waits where the road splits between route A and B until he either sees a train going from A to B or from B to A.
If the first train Steve sees goes from A to B he always takes route A and if the first train Steve sees is a train going from B to A he always takes route B.
There are equally as many trains going from A to B as there are going from B to A.
Nevertheless Steve has noticed that for every five times he visits his girlfriend he takes route A four times and route B only once.

How can this be?

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Reinout-g

edit: I updated a flaw in the illustration, 'work' was meant to be 'girlfriend'
Mar 22, 2014
 #5
avatar+2353 
0
radamus:
Phida:

How do you Solve this?
2(x+3) - 2x^2(x+3) = 0
In my answer sheet they rewrite it to (x+3) * (2-2x^2) = 0
Im unsure what happens to the other (x+3)
Thank you in advance !



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Simpler to do it differently than your answer sheet

Divide both sides of the equation by 2 and you get (don't forget 0 divided by anything is 0)
(x+3) - x^2(x+3) = 0

Divide by (x+3) and you get
1 - x^2 = 0

Divide by -1 and you get
-1 + x^2 = 0

Add 1 to both sides of the equation and get
0 + x^2 = 1

x^2 = 1
x = 1



Hey radamus,

Thanks for your help on the answer,
However, I do have two remarks on your answer.

Primarily, x 2 = 1 has two answers, both x = -1 and x = 1 since (-1) 2 = 1 is also correct

Secondly, you can't simply divide by (x+3) since for x = -3, you would be dividing by 0.
In your answer this means that the answer x = -3 was left out.

If you fill in x = -3 in the equation, you'll find that it is also correct.

Off-topic

To show you something fun which you can do by dividing by 0, here is a fallacious proof that 1 = 2

Step 1 Let a=b.
Step 2: Then a2 = ab,
Step 3: a2 + a2 = a2 + ab,
Step 4: 2a2 = a2 + ab,
Step 5: 2a2 - 2ab = a2 + ab - 2ab,
Step 6: and 2a2 - 2ab = a2 - ab
Step 7: This can be written as 2 ( a2 - ab) = a2 - ab,
Step 8: and cancelling the ( a2 - ab) from both sides gives 1=2.

Now in step 8, the mathematician divided by a2 - ab, but since a=b, a2 - ab = a2 - a2 = 0

That's why you should never divide by something which could be 0.

Reinout
Mar 22, 2014
 #1
avatar+2353 
0
jason Ullah:

Carbon is a major constituent of the Earth's composition. The term 'carbon cycle' is used to describe both the movement of this carbon through different reservoirs and the storage capacity of each reservoir. A considerable amount of this carbon is held in the Earth's biomass.

Table 1 provides estimates of the size of areas in square kilometres (km2) and the amount of biomass in kilograms of carbon (kgC) in different types of biome. A biome is composed of a characteristic range of vegetation that occurs within a distinct ecosystem type. It also provides estimates of the net primary productivity (NPP) for each type of biome in kilograms of carbon in each square metre (kgC m-2).
Table 1
Area Mean Plant Biomass Total Plant biomass Mean NPP Total NPP
Ecosystem Type (106 km2) (kgC m-2) (1012 kgC) (kgC m-2 yr-1) (1012 kgC yr-1)


Swamp, marsh 2 6.8 9.2 1.125 2.25
Lake, stream 2.5 0.01 0.01 0.225 0.563
Cultivated land 14 0.5 4.7 0.29 4.06
Rock, ice, sand 24 0.01 0.1 0.002 -
Desert, scrub 18 0.3 3.7 0.032 -
Tundra, alpine meadow 8 0.3 1.6 0.065 0.520
Temperate grassland 9 0.7 4.3 0.225 -
Woodland, shrubland 8 2.7 14.9 0.27 -
Savanna 15 1.8 18.3 0.315 4.73
Boreal forest 12 9 73.1 0.36 4.32
Temperate forest (deciduous) 7 13.5 64.3 0.54 -
Temperate forest (broadleaf, evergreen) 5 16 54.2 0.585 -
Tropical seasonal forest (deciduous) 7.5 16 81.3 0.675 -
Tropical rainforest (evergreen) 17 20 230.2 0.9 15.3

Total or Global mean 149 6.26 560 0.401 48.3

Use the data in Table 1 to answer the following questions. Note: if a question asks you to perform a calculation, make sure you clearly show your working as well as your answer.

a.Complete the final column of the table by calculating to 3 significant figures the total net primary productivity (NPP) for each type of biome (kgC yr-1).

(7 marks)
b.Rank in descending order the top five ecosystems (biomes) in terms of their area and state their area.

(1 marks)
c.Rank the top five biomes in descending order of their total annual net primary productivity (total NPP) and state their total NPP. Comment briefly on what your rankings in (b) and (c) show.

(3 marks)
d.Determine the percentage decrease in annual biomass production if 10% of the world's deciduous and tropical rainforest were cut down.

(4 marks)
e.Figure 1 is a summary diagram of the Earth's carbon cycle for modern times (i.e. post-Industrial Revolution).
Described image
View larger image
Figure 1 A summary of the movement of carbon in the world post-Industrial Revolution.
Long description

The mass of carbon in each reservoir is shown in gigatonnes (109 tonnes) which equals 1012kg. Thus the atmosphere contains 760 x 1012kg. The sizes of the fluxes between the reservoirs are given in units of 1012kgC yr-1. Where appropriate, the rates at which the processes of exchange take place are stated in italics, e.g., days for the flux between ocean and atmosphere; centuries for the flux between deep and surface ocean layers.

Compare the sizes of the present-day reservoirs (Figure 1) with those for the pre-industrial period (c. 1750 A.D.) given in U116 Block 4 (Figure 2.4). Indicate which reservoirs show change, and present this change as a percentage of the earlier value.




Please be aware that we try to help the users of this forum understand their questions and we do not in any way intend to do their homework for them.

Even though we might sometimes solve the question for them, we always attempt to give a complete and explainatory answer to them which we hope will help them understand the question next time.

Now if you have some serious trouble with the question, let us know where you're stuck with and why.

Even a try to what the correct answer may be is better than simply posting the question and hoping someone will answer.

I'm sorry if I sound a little harsh, but I want to be clear about what the difference is.

Reinout
Mar 12, 2014