The volume of the largest possible right circular cylinder that can be inscribed in a sphere of radius = 3 is:

Step-by-step solution

Given, radius of sphere = 3 Now, In OAB , by Pythagoras theorem ( OA )^2=( OB )^2+( AB )^2 aligned &( 3 )^2= ( h 2 )^2+r^2 &3= h^2 4 +r^2 r^2=3- h^2 4 aligned Now, volume of cylinder = r^2 h aligned &V= (3- h^2 4 ) h &V=3 h- h^3 4 aligned Now, for largest possible right circular cylinder the volume must be maximum For maximum volume, d V d h =0 Now, Differentiating eq. (2) w.r.t. hV^1= d V d h =3 - 3 4 h^2 or 3 - 3 4 h^2=0 3 = 3 4 h^2 h^2=4 h=2 Now, volume (V) of the cylinder = (3- h^2 4 ) h= (6-2)=4