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Question

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A silo is to be constructed in the form of a cylinder surmounted by

a hemisphere. The cost of construction per square unit of surface

area is 10 times as great for the hemisphere as it is for the

cylindrical sidewall. Determine the deminsions to be used if the

volume is fixed at 18000 cubic units and the cost of construction

is to be kept to a minimum. Neglect the thickness of the silo and

waste in construction

Explanation / Answer

Volume of hemisphere is just 1/2 that of a sphere so:
Vh = (1/2)[(4/3)pi*R^3] = (2/3)pi*R^3
where R is the radius of the sphere

Volume of the cylinder is just the base area times the height:
Area of base = pi*R^2
Height of cylinder = H
Vc = [pi*R^2]*H

Total volume = V = 18000
V = Vc + Vh = [(2/3)pi*R^3] + [pi*R^2]*H = 18000

Cost for construction.

Cost for cylindrial portion = C per unit area
Cost of hemisphere = Ch = 10C

Area of hemisphere is ne half the surface area of a sphere so:
Ah = (1/2)4*pi*R^2 = 2*pi*R^2

Area of cylinder (just the side, don't need top and bottom):
Ac = 2*pi*R*H

Total cost = M = Ah*Ch + Ac*C
M = (2*pi*R^2)(10C) + (2*pi*R*H)C

And you want to minimize M the total cost.

Use the volume equation to solve for H:
[(2/3)pi*R^3] + [pi*R^2]*H = 18000
H = {12000 - [(2/3)pi*R^3]}/[pi*R^2]

Put this into the equation for M and you will have an equation with just R as a variable (C is a constant and it doesn't really matter what it is except insofar as giving you the actual cost). Since we don't care what the total cost is I will just set C=1 so we can forget about it.

M = (2*pi*R^2)(10) + 2*pi*R*{18000 - [(2/3)pi*R^3]}/[pi*R^2]
M = (2*pi*R^2)(10) + 2*{18000 - [(2/3)pi*R^3]}/[R]
M = (2*pi*R^2)(10) + 2*{18000/(R) - [(2/3)*pi*R^2]}
dM/dR = 40*pi*R + 2*{-18000/(R^2) - [(4/3)*pi*R]}

Set this to 0 for the minimum

40*pi*R + 2*{-18000/(R^2) - [(4/3)*pi*R]} = 0
40*pi*R^3 - 36000 - (8/3))*pi*R^3 = 0
pi*R^3 - 900 - (1/15))*pi*R^3 = 0
R^3[pi - (1/15)*pi] = 900
R^3 = 900 / [pi(1 - 1/15)]
R = 8.835

Vh = (2/3)*pi*R^3 = (2/3)*pi*(8.835)^3
Vh = 1444.36

Vc = 18000 - Vh = 16555.5
Vc = [pi*R^2]*H = [pi*(5.89)^2]H
H = 16555.5/[pi*8.835)^2]
H = 67.5

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