The larger cone has a radius of 9 cm and a volume of 198pi cm^3. With these, let's determine the height of the cone using the formula of volume.
`V= 1/3 pir^2h`
Substitute the values of V and r.
`198pi = 1/3 pi (9)^2h`
To solve for h, divide both sides by `27pi` .
`22/3 = h`
So the height of the larger cone is 22/3 cm.
Since the two cones are similar, let's use ratio and proportion to solve for the height of the smaller cone.
Substitute `r_s=7.5 cm` , `r_L=9 cm` and `h_L=22/3 cm` .
`h_s/7.5 = (22/3)/9`
Then, multiply both sides by 7.5 to isolate `h_s` .
`7.5*h_s/7.5 = (22/27)*7.5`
So the height of the smaller cone is 6.11 cm.
Then substitute the radius and height of the smaller cone to the formula of volume.
`V = 1/3pi r^2 h= 1/3pi(7.5)^2(6.11) = 114.56pi`
Hence, the volume of the smaller cone is `114.56pi cm^3` .