# Take an open box, four sets of marbles (marbles in each set have the same size) and water. Place the marbles and water in the box and attempt the question- do larger marbles or smaller marbles...

Take an open box, four sets of marbles (marbles in each set have the same size) and water. Place the marbles and water in the box and attempt the question- do larger marbles or smaller marbles occupy more volume in a given space.(show the calculations also)

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In the experiment the question asks you to conduct, you would find that more water would be displaced (the water level would rise more) for the size of marble that occupies more volume. The question doesn't state that there are the same number of marbles in each set, but for ease of comparison we will assume that there are.

Intuitively, one would expect that larger marbles occupy more volume than do smaller marbles. Let us examine the math:

The volume of a sphere is as follows:

`V=4/3pir^3`

Therefore, as the radius of a marble increases, so too does the volume that it occupies. They are directly proportional (volume is a direct function of size [radius]). Our intuition is right, larger marbles occupy more space than smaller marbles. Two different sizes of marbles can raise the water level the same amount if there are more of the smaller marbles.

You would simply find the volume of each size of marble and multiply them by 6 to determine the total volume taken by each set of marbles. On the other hand, if you just want to know how much more volume one takes over the other, you just need to compare the cube of the radii: For example:

Let s = the radius of the smaller marble = 1cm

Let l = the radius of the larger marble = 2cm

`V_s=4/3pi(1)^3=4/3pi`

`V_l=4/3pi(2)^3=32/3pi`

Total volume taken by six smaller marbles = 6(4/3pi)=8pi

Total volume taken by six larger marbles = 6(32/3pi)=64pi

How many times the volume of the smaller marbles do the larger marbles multiply:

`(6(32/3pi))/(6(4/3pi))=(32/3pi)/(4/3pi)=32/4=8`

Therefore, the larger marbles take up 8 times more space than the smaller marbles.