When 8 spheres with a diameter of 1mm coalesce to form a single sphere how big will it be?

Answer 1

The volume of a sphere is 4/3 (pi) (R)3 .

The volume of each of the 8 spheres is 4/3 (pi) (0.5)3 .

That's (pi/6) mm3 each and 4pi/3 mm3 all together.

To find the size of the combined sphere, start again

with the volume formula and solve it for the radius:

V = 4/3 pi R3

Multiply each side by 3/4 :

3/4 V = pi R3

Divide each side by pi :

3V/4pi = R3

Take the cube root of each side:

R = the cube root of (3V/4pi)

Up above, we wound up with a volume of 4pi/3 mm3 .

So the radius of our new sphere is

cube root of (3V/4pi) = cube root of (3/4pi x 4pi/3) = cube root of (1) = 1 mm

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Check the answer (and a much easier way in which

the answer could have been found):

-- Volume of a sphere = 4/3 pi R3 .

Notice that the volume is proportional to the cube of the radius,

and therefore also to the cube of the diameter.

-- The volume of the big sphere was 8 times the volume of the little ones.

So the big diameter should be the cube root of 8 times the small diameter.

-- The cube root of 8 is 2.

The small diameter is 1.

The big diameter should be 2 x 1 = 2 .

-- I calculated the big radius as 1.

-- Yay !