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
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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
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
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