Question

prove that the radius of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half of that of the cone

Answer 1

H = height of the cone , R = radius of the cone 
h = height of the cylinder, r = radius of the cylinder, 
S = lateral surface area of the cylinder 

S = 2π r h 
r = R (1 - h/H) .

S = 2π (R (1 - h/H)) h 
dS/dh = 2π (R/H) (H - 2h) 
d²S/dh² = -4π (R/H) 
H - 2h = 0 .......... set dS/dh = 0 to find the stationary points 
h = H/2 .............. as d²S/dh² < 0 

r = R (1 - (H/2)/H) ....... plug h=H/2 into r = R (1 - h/H) 
r = R/2