Question

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?

Answer 1

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

Answer 2

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

Answer 3

Yes, of course, sorry