A silo base not included is to be constructed in the form of

A silo (base not included) 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 dimensions to be used if the volume is fixed at 14000 cubic units and the cost of construction is to be kept to a minimum. Neglect the thickness of the silo and waste in construction. The radius of the cylindrical base (and of the hemisphere) is ft.

Solution

We will use the following formulae.

Surface area of a cylinder is 2rh +2r²

Volume of cylinder = r²h

Volume of a sphere = 4/3 r³

Surface area of a sphere = 4r².

Let r be the radius of the base of the cylinder and h be its height. Then the radius of the hemispere is also r. The volume of the cylinder is r²h and the volume of the hemisphere is (½)*4/3 r³ = 2/3 r³. Then the total volume of the cylinder and the hemisphere is 2/3 r³+ r²h = 14000 or, r²h= 14000 – (2/3 ) r³ so that h = 14000/r² -2r/3. Also, the surface area of the cylinder ( ignoring its base and the top) is 2rh. Let the cost of construction for the cylinder be x per square unit . Then the cost of construction for the hemisphere is 10 x per square unit so that the total cost of construction is            x(2 rh) + 10x(2r²) = 2xrh + 20 xr² = 2xr(14000/r² -2r/3) + 20 xr²= (28000x/r ) - 4xr²/3 + 20 xr² =   (28000x/r ) + (56/3 ) xr² = C(r) (say).

Then dC/dr = (112/3) xr - 28000x/r² and d²C/dr² = (112x/3)+ 28000x*2/r³ =(112x/3)+ 56000x/r³   

If C(r) is to be minimum, then dC/dr = 0, and d²C/dr² should be positive so that = 0 or, (112/3) xr - 28000x/r²   = 0 or,28000= (112/3) r³ so that r = (28000*3/112)^1/3 = (750/)^1/3. Further, it may be observed that d²C/dr² is positive regardless of the value of r.

Thus, if the cost of construction is to be kept minimum, then the radius of the base of the cylinder (and of the hemisphere) is (750/)^1/3 = ( 750/3.14159)^1/3 = (238.73)^1/3 = 6.20 units (approximately).