Silo Calculus Question?
Can anyone help me out with this calculus question
A silo must be designed to hold pi * 10,000 ft^3. The silos are to be right circular cylinders with a hemispheric roof. The sheet metal that the walls of the cylinder must be constructed with costs $1/ft^2. The roofing material costs $3/ft^2 and the flooring costs $2/ft^2. Find the dimensions of the storage silo that will minimize construction costs.
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v=10000pi=2/3pir³+pir²h
dr/dh=(10000/r²)-(2r/3)
cost=2pirh+2pir²+6pir²
=2pi(rh+4r²)
using volume, find h=10000/r²-2r/3
plug into cost formula and do c'
=8pir-20000/r²
c'=0 for crit points
r=9.267
For dimensions:radius is 9.267, height of cyclinder is 110.27
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v=10000pi=2/3pir³+pir²h
dr/dh=(10000/r²)-(2r/3)
cost=2pirh+2pir²+6pir²
=2pi(rh+4r²)
using volume, find h=10000/r²-2r/3
plug into cost formula and do c'
=8pir-20000/r²
c'=0 for crit points
r=9.267
For dimensions:radius is 9.267, height of cyclinder is 110.27
References :