Question

Circle problem

Answer 1

\[\left[ \frac{120}{\pi}\right]\]Probably.

Answer 2

lol

Answer 3

what is that????

Answer 4

I think I did something wrong, is my representation for greatest integer function wrong?

Answer 5

Or the Floor Function

Answer 6

Sorry... I don't understand :(

Answer 7

Yes it is, don't even think about understanding it.

Answer 8

Wrong guess Ishaan, 38 isn't a option.

Answer 9

Is the sheet is still intact or it is teared like the figure?

Answer 10

Hmm... Not sure what you're asking. But the sheet is 40cm long

Answer 11

I think as it's a metal sheet broken is the right word.

Answer 12

The sheet appears broken.

Answer 13

Actually .. you don't have to consider if it is broken...

Answer 14

20? Maybe 20 isn't right either

40/2.

Answer 15

I did it in this way:
(40-2)/sqrt3 +1 = 22.9, round down => 22

Answer 16

Hmm.. the answer is C, if the answer is correct :S

Answer 17

But why you did like that?

Answer 18

@Callisto

Answer 19

Wait... some problems...

Answer 20

1 + n sqrt3 + 1 = 40
n sqrt3 = 38
22.

Answer 21

But I'm not sure as always

Answer 22

What bugs me I wasn't able to think about that sqrt3 part. :/

Answer 23

And gave a foolish reply at the start :(

Answer 24

It should be 22.

Answer 25

40 - radius of 1st circle - radius of last circle
then, divided by sqrt3
round down the number
as you've minus a diameter, you need to add it back (that is to add one 1 more circle)
which gives 22,
Not sure if the explanation is correct

Answer 26

Well what I would do is see what is the length of the position of the first two balls

Answer 27

Why do you have to add 1 circle back? @Callisto

Answer 28

''as you've minus a diameter, you need to add it back (that is to add one 1 more circle)''

Answer 29

so the diagonal of the first two circle (2* r) + ((squareroot of 2) - 1)

Answer 30

use pythagurus to find the unknown side (Length) since you already know the width to be 3 cm

Answer 31

The only thing I can be sure of in the question is to add 1 in this question, as my teacher has said it. I was not quite sure of the calculation part

Answer 32

once you find the side see how many fit on the sheet in regards to the length

Answer 33

That's weird. You have minus 2 radius because you want to make sure that the first and last circle is intact. I think by adding 1 circle back will make a circle that is not intact. So my answer is 21.

Answer 34

I got 24 actually so I think I did it wron

Answer 35

@thomas5267 my explanation is not clear though. but you need to know that the first one must be counted but you've neglected it in the calculation.. Just consider the first 2 circle. if you just divide sth by sqrt 3, you'll have 1 as the answer, but actually you have 2 circles. Based on similar situation, you need to add 1 back to the answer you get. Hope that you understand

Answer 36

I'm thinking the answer shuold be given by \[\left\lfloor {38 \over \sqrt{3}}\right\rfloor=21\]

Answer 37

Nice, so my foolish wasn't that foolish.

Answer 38

@Ishaan94 what do you mean?

Answer 39

To add, or not to add, that is the question.

Answer 40

I liked @Ishaan94's second answer. It should be 22.

Answer 41

I understand your statement but I can't figure out when have we neglected a circle. @Callisto

Answer 42

It is 22, but not through rounding up. The question stated that 'at most' 21.9 doesn't mean that it can form 22 circles ...

Answer 43

20 + 10(2 - sqrt3) = [20 + 10(2-sqrt3)]

Answer 44

The trick is to find the x-coordinate of the second circle. You know the first circle is centered at \(1, 2\), and the next one at \((x, 1)\) for some x. Since they both have radius 1, and are supposed to be tangent, yet minimizing some other things, you know that the line from the centers must have slope of -1. This implies the circles are tangent at \(y=3/2\) and \(x=1+\sqrt{3}/2\). Thus the distance between the x-coordinates of the first two circles is \(\sqrt{3}\).

By subtracting off of a radius of a circle at either end, we have the x-coordinates of the first and last circles fixed at \(x=1, 39\). Divide by \(\sqrt{3}\) and round down, and we get the number of centers between \(x=1, 39\) excluding \(x=39\). Thus we have to add one back in to get 22.

Answer 45

the only reasonable answer I see is from KingGeorge :)

Answer 46

King has to be reasonable.

Answer 47

Wouldn't have seen the adding back in unless Fool mentioned he liked 22 better.

Answer 48

The answer is 22. The center of the last red circle is \[ 1+22 \sqrt3\]
so \[ 1+1+22 \sqrt3=40.1051\]
so it does not fit.