A steel storage tank for propane gas is to be designed in the shape of a cylinde

Question

A steel storage tank for propane gas is to be designed in the shape of a cylinder with a hemisphere at one end and a Hat circular end at the other as shown in the figure below. The construction cost to fabricate each portion of the tank is given below. In addition to the fabrication costs above to form each part of the tank into the appropriate shape, welds are used to join each piece of the tank together. A weld is placed completely around each end of the cylindrical portion of the tank to connect the ends of the tank as shown in the figure below. In addition, a weld runs the entire length of the inner cylindrical portion of the tank. The black lines shown in the diagram below arc welds that cost $100 per foot. If the tank is to be designed to have a volume of 300 cubic feet, determine the dimensions of the tank that will minimize its cost of construction.

Explanation / Answer

Solution:

All we need to find is the mimimum surface area,

because the cost of the construction depends of the surface area.

given the volume is 300 ft^3

300 ft^3 is the volume of the right circular cylinder plus the volume of
hemisphere at each end and flat circle at other end

the radius of the circular cylinder is also the radius of the hemispheres

There are hemisphere and cylinder
so,

V = volume of the right circular cylinder + volume of the hemisphere

V = r² h + ( 2/3 r^3 )

V = r² h + 2/3 r^3

300 = r² h + 2/3 r^3

300 - 2/3 r^3 = r² h

h = (300 - 2/3 r^3) / ( r²) .... 1

Surface Area = 2 r h + 2 r² + r²

substitute h from 1

Surface Area = 2 r (300 - 4/3 r^3) / ( r²) + 3 r²

SA = 2 r (300 - 2/3 r^3) / ( r²) + 3 r²

on differentiating SA with respect r
we get

d/dr (SA)= (2r^3 -1800)/3r^2

for minimum surface area
d/dr(SA) = 0
(2r^3 -1800)/3r^2 =0
we get
real value r = 3.85 = 3.85

as we calculate h , we get

height h = 3.85

so when radius and heigh both are = 3.85 at this point the construction cost can be minimized

Answer

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