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Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base.

Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base.
 

Grade:12

1 Answers

Arun
25763 Points
2 years ago

S is the total surface area of the closed right circular cylinder whose radius is r, height is h, and volume is V.
Then surface area S = 2 * pi * r* h + pi * r^2 * h

 ------ (i)
V = ½ [Sr – 2*pi*r^3]

Therefore, dV / dr = ½ [S – 6 * pi * r^2 ]

For extrema (maxima or minima) dV/dr=0

Therefore S – 6 * pi * r^2 = 0 
S – 6 * pi * r^2 = 0
Plugging this value of S in eq.(i),we get

h (at V-extrema)= 2r
For maxima, d2V / dr2 = – 6* pi*r = negative 

Therefore, V is maximum when height, h = 2r = diameter of the base.

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