Posted by **Xian** on Thursday, December 26, 2013 at 5:38pm.

The surface area and the volume of a sphere are both 3 digit integers times π. If r is the radius of the sphere, how many integral values can be found for r?

- Math -
**Damon**, Thursday, December 26, 2013 at 5:57pm
4 pi r^2 = (xyz)pi

(4/3) pi r^3 = (ijk) pi

4 r^2 = xyz

(4/3) r^3 = ijk

(4/3) r^3 is an integer less than 1000

and greater than 99

if (4/3) r^3 = 1000 r is 9.08 so try 9

if r = 9, (4/3)r^3 = 972 , the biggest we can have for r^3 criterion

now if (4/3) r^3 = 100 r is 4.21

so smallest r for this criterion is

r = 5

so far we have

5 , 6, 7, 8 , 9

BUT we need to do the area thing

4 r^2 = xyz

if xyz = 1000, r = 15.8

So

15 is the biggest for the area

if xyz = 100, r = 5 exactly

so between 5 and 15

so

5,6,7,8,9 seem to work, five of them

- Math -
**Steve**, Thursday, December 26, 2013 at 7:37pm
But 4/3 r^3 must be an integer, so r must be a multiple of 3.

I'd say only 6 and 9 are available

- Math -
**Damon**, Thursday, December 26, 2013 at 9:25pm
Yes, of course, sorry

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