What is the largest number of balls of radius R that can be fitted

Question

What is the largest number of balls of radius R that can be fitted (without compressing) inside a rectangular box of internal dimensions 2R, 3R, and 9R?

A. 4
B. 5
C. 6
D. 12
E. 54

chetan2u · Answer

You would go wrong if you believe 2*3*9 will be exactly like 2*2*9. Since, you can place the balls side by side, the height from center to center of two adjoining balls will go below 2R. Look at the figure attached. BALLS IN CYLINDER.png The distance between the center of the two balls is R+R or 2R, also given by BC. AB is the horizontal distance between the two balls = 3R-R-R or R Thus, the vertical distance or height = AC = \sqrt{(2R)^2-R^2}=\sqrt{3}R=1.73R Since we have a height of 9R available with us, if x is the number of balls that can fit in, x-1 will have a vertical distance of 1.73R => R+1.73R*(x-1)+R\leq 9R........1.73(x-1)\leq 9-2............x-1\leq \frac{7}{1.73}......x-1\leq 4.04.......x\leq{5.04} Therefore, max number can be 5. B

bethebest96 · Answer

ThatDudeKnows  plz hlp .

ThatDudeKnows · Answer

bethebest96

First thing I did was get rid of R. We can treat the question like this: What is the largest number of balls of radius 1' that can fit inside a rectangular box of internal dimensions 2'x3'x9'.

Next, let's visualize this box as lying flat on one of the largest faces, such that the height is 2'. Since each ball has a radius of 1', height is 2', and we therefore know that each ball will be touching both the bottom and top of the box. Aha, we can now rephrase the question once again: What is the largest number of CIRCLES of radius 1' that can fit inside a   rectangle of dimensions 3'x9'. That just feels like an easier question to visualize/draw.

Draw the rectangle and draw in your first circle all the way to one corner. Since the radius is 1' and diameter 2', the space remaining under the circle is 1'.

Draw in the second circle, squeezing it as far to the left as it'll go. What do we know about the figure now? I can describe it in words, but look at the attachment and you'll likely be able to start seeing what's going on.

How many circles can we add and still fit within the limit of 9'? Well, the distance from the left edge of the rectangle to the center of the leftmost circle will always be 1. And the distance from the center of the rightmost circle to the rightmost edge of that same circle will also be 1'. And then the horizontal distance between any two circles will be  \sqrt{3} .

Now, how many circles can we fit? We could come up with a formula, but the numbers are small enough, so let's just examine how wide the box would have to be given a certain number of circles.
1 circle: 2'
2 circles:  2+\sqrt{3}  = 3.7'
3 circles:  2+2\sqrt{3}  = 5.4'
4 circles:  2+3\sqrt{3}  = 7.1'
5 circles:  2+4\sqrt{3}  = 8.8'

We clearly don't have enough space to add a sixth.

Five balls.

Answer choice B.

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What is the largest number of balls of radius R that can be fitted

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What is the largest number of balls of radius R that can be fitted

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