Please Help,hard mathematical question,smart people only!?

A conference centre contains five identical rooms. All of the rooms have square floors. Recently, the floors of the rooms have been painted, and a different pattern has been used for each room. Each room has a certain number of red circles painted on a white background in the patterns
Room 1 has 9 red circles of equal area painted on a white background
Room 2 has 16 red circles of equal area painted on a white background;
Room 3 has 25 red circles of equal area painted on a white background;
Room 4 has 36 red circles of equal area painted on a white background;
Room 5 has 49 red circles of equal area painted on a white background;

Which room has the largest red-painted area on its floor? If one room has a greater red-painted area than the rest, give the number of that room (between 1 and 5 inclusive) as your answer. If all of the rooms have the same red-painted area, give 0 as your answer.

room 5

We need some more information, like the fact the the circles are touching, in a regular square grid?

If they are, then a circle inside a square takes up 78.540% of the area, no matter what the size of the square is. So the answer is, all the rooms have the same red area.

But, if the circles are arranged is some other pattern, like a close-packed grid, then the answer might be different.

5

It depends on how the circles are arranged on the floor. It also depends on whether the circles painted on the floor are the largest possible circles that will fit on the floor.

Please clarify your question.

The numbers you picked are all perfect squares. If the circles are touching and arranged in a square grid then it doesn't matter. All of the rooms will have the same area painted red.

its impossible bc the room with 9 circles could have circles the same size as the room with 49 circles or it could have circles 10x them. in other words you would have to know the size of the circles, wouldnt you?

Each room is 1 unit * 1 unit

Using the assumption that the circles are arranged in a square, all circles in a room are the same size and the circles are touching without overlapping.

Room 1
Diameter of 1 circle = 1/3
Area = 1/6虏蟺 = 1/36蟺
9 circles = 9/36蟺 = 1/4蟺

Room 2
Diameter of 1 circle = 1/4
Area = 1/8虏蟺 = 1/64蟺
16 circles = 16/64蟺 = 1/4蟺

Room 3
Diameter of 1 circle = 1/5
Area = 1/10虏蟺 = 1/100蟺
25 circles = 25/100蟺 = 1/4蟺

Room 4
Diameter of 1 circle = 1/6
Area = 1/12虏蟺 = 1/144蟺
36 circles = 36/144蟺 = 1/4蟺

Room 5
Diameter of 1 circle = 1/7
Area = 1/14虏蟺 = 1/196蟺
49 circles = 49/196蟺 = 1/4蟺

All rooms have the same total area of red circles.

Ans: 0

Assuming that the circles all touch and there are the same number of rows as columns of circles, all the rooms have the same red painted area. This is the assumption I'll make and my answer is 0.

Fun question...

Best thing to assume is a standard square room and then calculate the area of the circles from the total area of the square.

If you wanted to do the calculations for this instead, assume a 10 x 10 room for 100 square feet.

in the first room, the circles would have a radius of 10/3/2 (3 circles next to each other, radius is half of one circle).

Therefore the area of one circle is pi*r^2. = 8.73 * 9 circles = 78.5 square feet of the floor is painted.

Now, do the largest number of circles... (49 or 7 by 7). Each circles radius would be 10/7/2 (10 divided by 7 circles divided by 2 for the radius). or .7143.

Same equation this time. pi*r^2 = 1.604 * 49 circles. 78.5 square feet of the floor is painted.

You can do the same equation for the rest, but you will find the same area of square footage painted in each scenerio. You would think that there would be more area painted with more circles, but more circles create more smaller areas of white unpainted space as they are jammed in together.

In summary, your answer is 0.

Due to the fact that the white area remaining in the 9 circle room is large, but off set by the lack of white spaces intra-circularly, then that proportion will probably carry through each room increasing with the red space equaling the white space lost. I'm going to guess that they would all be the same in their red space as the white space gained with increasing number of circles would balance out the red space gained by the increasing dots.

My answer is 0 because red spaces are equal in all rooms.
GREAT QUESTION!