Find the radius of the cylinder that produces the minimum surface area.

Question

A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 18 cubic centimeters. Find the radius of the cylinder that produces the minimum surface area. (Round your answer to three decimal places.)

Answer 1

radius of cylinder and two hemisphere = r
Volume of 2 hemispheres = Volume of 1 sphere

V₁ = 4/3 π r³

Volume of cylinder
V₂ = π r² h
Total volume of solid = 18 cm³
4/3 π r³ + π r² h = 18
π r² h = 18 - 4/3 π r³
h = (18 - 4/3 π r³) / (π r²)
h = 18/(π r²) - 4/3 r                            

Surface area of 2 hemispheres

S₁= Surface area of 1 sphere  = 4 π r²

Lateral surface area of cylinder, S₂ = 2 π r h

Total surface area , S = S₁ + S₂

= 4 π r² + 2 π r h

= 2 π r(2r + h )

Substitute h = 18/(π r²) - 4/3 r  

S = 2 π r (2r + 18/(π r²) - 4/3 r)
  = 2 π r (2/3 r+ 18/(π r²))

  = (4π/3 )r² + 36 /r

Find the radius of the cylinder that produces the minimum surface area
S' = 8r/3 π - 36/r²

S' = 0

8πr/3  - 36/r² = 0
36/r²  = 8πr/3

8πr³ = 108

r³ = 4.3

r = 1.63 cm

S'' = 8/3 π + 72/r³

   > 0

Thus the radius of the cylinder that produces the minimum surface area is 1.63 cm

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