Question

A given quantity of metal is to be cast into a solid half circular cylinder with

a rectangular base and semi-circular ends. Show that in order that total

surface area is minimum, the ratio of length of cylinder to the diameter of

semi-circular ends is π ∶ π + 2.

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Answer 1

Step-by-step explanation:

Given A given quantity of metal is to be cast into a solid half circular cylinder with  a rectangular base and semi-circular ends. Show that in order that total  surface area is minimum, the ratio of length of cylinder to the diameter of  semi-circular ends is π ∶ π + 2.

• Let r be the radius and h be the height of half cylinder.
• Let V and S be total volume and total surface area of half cylinder.
• Volume of half cylinder = 1/2 x πr^2 h
• So 2v / π = r^2 h
• Or r^2h = 2v / π
• Or h = k / r^2 where k = 2v / π
• Now total surface of half cylinder = 1/2 x curved surface area of cylinder + 2 x area of semicircle + area of bottom rectangle.
• =   1/2 x 2πrh + 2 x (1/2 πr^2) + 2r h
• = πrh + πr^2 + 2rh
• Putting h = k/r2
• = πr (k/r^2) + πr^2 + 2r x k/r^2
• = π x k/r + πr^2 + 2k / r  
• = π k x 1/r + πr^2 + 2k x 1/r
• Differentiate w.r.t r we get
• So ds/dr = πk x (1/r)’ + π (r^2)’ + 2k (1/r)’
•              = πk x (-1/r^2) + π x 2rn+ 2k x (-1 / r^2)
•              = - πk x 1/r^2 + 2π x r – 2k x (1/r^2)
• Put  ds/dr = 0
•        0 = - πk x 1/r^2 + 2π x r – 2k x (1/r^2)
•     So 2πr = 2k x (1/r^2) + πk x 1/r^2
•     2πr = k(2/r^2 + π/r^2)
•    2πr = k(2 + π / r^2)
• So k = 2πr^3 / (2 + π)
• Now ds/dr = - πk x 1/r^2 + 2 π x r – 2k (1/r^2)
• So ds/dr = k(-π / r^2 – 2/r^2) + 2 πr
• So ds/dr = -k(π + 2/2) x 1/r^2 + 2 πr
• Differentiate again w.r.t x we get
• So d^2 s / dr^2 = - k(π + 2/2) x – 2/ r^3 + 2 π
• Now d^2s / dr^2 = k(π + 2) x 1/r^3 + 2 π
• Therefore d^2s / dr^2 > 0 when k = 2 πr^3 / (2 + π)
• So s is minimum when k = 2 πr^3 / (2 + π)
• Now 2 πr^3 / (2 + π) = 2V / π
• Volume = 1/2 πr^2 π  
•  2 πr^3 / (2 + π) = 2 x 1/2 πr^2 h / π
•                = πr^2 h / π
•                           = r^2h
• Now r^3 / r^2 h = (2 + π) / 2π
• Or h/r = 2π/ (2 + π)  

Therefore h/2r = 2π / (π + 2)