Calculus
posted by Rae
A steel storage tank for propane gas is to be constructed in the shape of a right circular cylinder with a hempishere at each end. The cost per square foot of constructing the end pieces is twice that of contructing the cylindrical piece. If the desired capacity of the tank is 144 pi cubic feet, what dimensions will minimize the cost of construction?

Reiny
The two hemispheres at the ends will make up one complete sphere
Let the radius of the sphere be r ft, and let the height of the cylindrical part be h
Volume = cylinder + sphere
πr^2 h + (4/3)πr^3 = 144π
3πr^2 h + 4πr^3 = 432π
3r^2 h + 4r^3 = 432
h = (432  4r^3)/(3r^2)
The cost of production is dependent on the surface area of material used
SA = 4π^2 + 2πrh
= 4πr^2 + 2π(432  4r^3)/(3r^2)
= 4πr^2 + 288π/r^2  (8/3)πr
d(SA)/dr = 8πr  576π/r^3  8π/3 = 0 for a min of SA
8r  576/r^3  8/3 = 0
times 3r^3
24r^4 1728  8r^3 = 0
3r^4  r^3  216 = 0
At this point I "cheated" a bit and ran it through Wolfram
http://www.wolframalpha.com/input/?i=3r%5E4++r%5E3++216+%3D+0
for a real solution of r=3, r = appr3 and 2 complex roots,
so the only usable answer is r = 3
the h = (432  108)/27 = 12
So the cylinder part should be 12 ft long and the radius of each of the semispheres should be 3 ft 
Damon
remember the sphere surface is twice as expensive as the cylinder which has no compound curvature so is easy to roll.
c = 2(4 pi r^2) + 1(2 pi r h)
c= 8 pi r^2 + 2 pi r h
Respond to this Question
Similar Questions

geometry
A propane tank is in the form of a cylinder with two halfspheres, one on each end. The radius of the cylinder is the same as the radius of each halfsphere. What is the volume of a propane tank of cylindrical length 12 ft and radius … 
the lagrange multiplier equation
A cylindrical oilstorage tank is to be constructed for which the following costs apply: cost per square meter metal for sides $30.00 (cost per square meter) combined costs of concrete base and metal bottom $37.50(cost per square meter) … 
mathoptimal diameterand height
A cylindrical oilstorage tank is to be constructed for which the following costs apply: cost per square meter metal for ides $30.00 combined costs of concrete base and metal bottom $37.50(cost per square meter) top 7.50 (cost per … 
calculus
A tank with a rectangular base and rectangular sides is to be open at the top. It is to be constructed so that its width is 4 meters and its volume is 36 cubic meters. Building the tank costs $10 per square meter for the base and $5 … 
maths
A storage tank is in the shape of a right circular cylinder that has a diameter of 20 inches and a height of 3yards. if oil costs $2 per cubic foot, what does it cost to fill the storage tank oil? 
college algebra
A propane tank has the shape of a circular cylinder with a hemisphere at each end. The cylinder is 6 feet long and volume of the tank is 5pie cubic feet. Find, to the nearest thousandth of a foot the length of the radius x. 
lagrange multipliers
A cylindrical oilstorage tank is to be constructed for which the following costs apply: cost per square meter metal for ides $30.00 combined costs of concrete base and metal bottom $37.50(cost per square meter) top 7.50 (cost per … 
Calculus
An energy drink container in the shape of a right circular cylinder must have a volume of 12 fluid ounces (1 fluid ounce is approximately 1.80469 cubic inches). The cost per square inch of constructing the top and bottom is twice the … 
Calculus
A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. An industrial tank of this shape must have a volume of 1600 cubic feet. The hemispherical ends cost twice as much per square foot of surface area … 
Calculus
A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. An industrial tank of this shape must have a volume of 1600 cubic feet. The hemispherical ends cost twice as much per square foot of surface area …