Question

A 1000 gallon (133.7ft^3) propane tank is to be constructed as cylinder with two hemispherical ends. The design is constrained such that the total length from one end to the other is twice the diameter. Determine the following:

- The diameter of the tank in feet

- The overall length of the tank

- The volume of steel required if the tank wall is 1/4 - thick. Note that the volume of the walls can be calculated by multiplying the total surface area of the tank by the thickness.

Answer 1

Given data:

Total volume: 133.7 ft^3

now, the tank to be constructed must be of cylinder with two hemispherical ends

let 'r' be the radius of the cylinder and the hemispere

'd' be the daimeter

according to the condition total length =2d

volume of the new tank = 2(volume of hemisphere) + volume of the cylinder

volume of hemisphere = 2/3 *pi* r^3

volume of cylinder= pi*r^2* l   [here height= length]

as l= 2d= 2*2r=4r

volume of the new tank= 2(2/3 *pi*r^3) + (pi* r^2 *4r)

                                 =4/3 * pi* r^3    +   4 pi * r^3

                                  =4 * pi * r^3( 1/3 + 1)

                                  =16/3 * pi * r^3

now total volume = new tank volume

133.7= 16/3 * pi * r^3

[ here pi =22/7 ]

r^3=7.97642

r = 1.99 feet

therefore radius r= 1.99 feet

           length= 2d= 4r= 7.96

volume of the steel required= surface area of the tank * width of the wall

given that width of the wall = 1/4 feet

total surface area of the tank = 2( surface area of the hemisphere ) + surface area of the cylinder with open lids

                                          =2 ( 2 * pi * r^2) + 2 * pi * r* l

                                          = 4 * pi * r^2 + 8* pi * r^2 [ since l=4r]

                                         = 12* pi * r^2

                                         =12* 22/7 * (1.99)^2

                                         =149.3523 feet^2

total steel required = thickness * surface area = 1/4 * 149.3523

                                                                     =37.33809 feet^2