Problem Solving

How many perfectly spherical balloons of 1feet radius can fit in a room that is 10 feet wide by 12 feet long by 8 feet high .Without compressing any of the balloons or allowing balloon to touch more than 6 other balloons.
a(30
b)120
c)144
d)360
E)960

IMO A.

without the resitiriction, Without allowing balloon to touch more than 6 other balloons.

the number of balloons that can fit in the room is

since balloons are 2x2ft and thinking them as a cube,
10/2*12/2*8/2=120

applying the restriction, the number has to be less than 120, only choice less than 120 is A.

but I'm not sure to think balloons as a cube is probably a flaw

is there an OA?

answer is 120

wait...

does "Without allowing balloon to touch more than 6 other balloons. " mean that balloon is allowed to touch 6 other balloons?....no? I dunno

even if that was true, i have no idea how to prove why any number below 230 is wrong.

Someone help XD

jagmohi wrote:How many perfectly spherical balloons of 1feet radius can fit in a room that is 10 feet wide by 12 feet long by 8 feet high .Without compressing any of the balloons or allowing balloon to touch more than 6 other balloons.
a(30
b)120
c)144
d)360
E)960

My educated guess - I don't know how to solve the final restriction.

So 10X12X8 = 960

The balloon volume is (4/3)(PIE)(R^3)

This ends up being 4/3 Pie

4/3 Pie approximates to 4 cubed feet.

960 / 4 = 240

Eliminate D/E

A is too extreme

120 / 144.

Somehow I chose B. 240/2 perhaps. I dunno. I got it down to 50/50 and just guessed.

Answer is 120, it is correct to look at the balloons as cubes.

The fact that they cannot touch more than 6 balloons refers to the fact that there are 2 possible confiurations to make them fit in a room:
like this with 6 touch points
OOOO
OOOO
or like this with 8 touch point (sorry for the bad graphic)
O O O O
O O O

with the second one (more complicated to calculate) more balloons fit.

Max no of balloons in the room can be 210.

Hence one can eliminate 2 options from the given options and obviously 1st option so there lies a chance btn 2nd and 3rd option..

I dont know how to find the exact answer.

Correct ans is "120" So "B"
Here is the explaination, visualise the arrangement, its simple....
In centre
o
o o o
o

and one ballon above the center another below the center.
So max 6 will touch.
Now 12/2*10/2*8/2=120
Is it clear?

How can you say if that is the only arrangement possible??? Can you be bit clear about this?

Answer is 120.

Since the dia = 2
Length 12: you can have 6 max.
Width 10: you can have 5 max.
Height 8: You can have 4 max. Lets call them 4 levels.
Now here lies the key. The point of touching 6 max balloons indicates that the balloons must be placed right on top of each other. So for example, the center of the balloon on second level should line up with the center of the balloon on level 1 as well as level 3. Makes sense? The other arrangement refers to the fact that say if you have 4 balloons on level 1 arranged in a square so all touching the neighboring balloon, the level 2 balloon could be placed in the center of these 4 and hence touches all 4 of them. But if you placed it exactly on top of one of the balloons, it would touch only 1 balloon and not 4.

Let me try to display it...

WRONG arrangement: Here the balloons in level 2 can touch upto 8 balloons. 4 from level 1 and 4 from level 3.

Level 4: OOOO
Level 3:OOOOO
Level 2: OOOO
Level 1:OOOOO

Right arrangement with the restriction of of not touching more than 6 balloons:
Level 4:OOOO
Level 3:OOOO
Level 2:OOOO
Level 1:OOOO


So the answer is 6* 5 * 4 = 120.

Hope that helps.