It'd be cool of someone could answer this soon, thanks!

See the following ACG :

Since the sphere is tangent to  both the top and bottom bases, the height of the frustum must be   = 6  = BE

And the slant height   = 10  = BD

Notice that  since  AB   and BC  are tangents drawn from  the same point, they are equal

And AB  = 1/2 ot the top  base  of the frustum.....so BC  equals the same.... and  call this, m

But  CD  and  DF are also  tangents drawn from the same point.....so CD  = DF

And because  of the Pythagorean Theorem,  DE  = √ [BD^2  - BE^2=  = √[10^2 - 6^2 ]  =   8  =   r  - m

So.....we have the following eqautions

r + m  = 10

r - m  =   8      add these

2r  = 18

r  = 9   =  bottom radius

So...  9 + m  = 10  ⇒ m  = 1   = top radius

So....the  volume of the frustum  is

(1/3)  pi  *  height  * ( top radius^2  + product of the radii + radius ^2 )  =

(1/3) pi  *  6  * ( 1^2  +  1*9   +  9^2 )  =

(1/3) pi * 6 * ( 1 + 9  + 81)  =

(1/3) pi * 6 ( 81)  =

182 pi  units^3

Edited to correct an error  !!!

Can you copy and paste the other solution so that they can be compared to see where the mistake is?