`a)` `r=\sqrt{R^2-(h/2)^2}=2\sqrt{5} (cm)`

`b)` `V_(\text{trụ})=\pir^2h=160\pi (cm^3)`

`V_(\text{nón})=(1)/(3)\pir_(\text{nón})^2h=(1)/(12)\pir^2h=(40\pi)/(3) (cm^3)`

`c)` `V_(\text{cầu})=(4)/(3)\piR^3=288\pi (cm^3)`

`=> (V_(\text{nón}))/(V_(\text{cầu}))=(5)/(108)`

`d)` Có: `r^2=R^2-(h/2)^2=36-(h^2)/(4) (cm)`

`(h in (0; 12))`

Có `V_(\text{nón})=(1)/(3)\pir_(\text{nón})^2h`

`=(\pi)/(3).(1)/(4).(36-(h^2)/(4))h`

`=(\pi)/(3).(9h-(h^3)/(16))`

Xét `V'=(\pi)/(3).(9-(3h^2)/(16))=0`

`=> h=4\sqrt{3} (cm)`

Bảng biến thiên:

$\begin{array}{c | ccc} x & 0 & & 4\sqrt{3} & & 12 & \\\hline V' & & + & 0 & - & \\\hline V & & & 8\sqrt{3}\pi &\\ & & \nearrow & & \searrow &\\  & & \end{array}$

`=> Max_(V_(\text{nón}))=8\sqrt{3}\pi cm^3` khi

`h=4\sqrt{3} cm`