Project Setup:You work for a company that manufactures cylindrical steel drums that can be used to transport various petroleum products. Your assignment is to determine the dimensions (radius and height) of a drum that is to have a volume of 1 cubic meter while minimizing the cost of the drum.You will do this in stages, calculating the optimal dimensions for the part of the situation described at each stage.
1.The cost of the steel used in making the drum is $3 per square meter. The top and bottom of the drum are cut from squares, and all the unused material from these squares is waste. The remainder of the drum is formed from a rectangular sheet of steel, assuming no waste. Ignoring all costs other than material cost, find the dimensions of the drum to minimizethe cost of materials(steel).
2.The drum has seams around the perimeter of the top and bottom, as well as a vertical seam where the edges of the rectangular sheet are joined to form the lateral surface. In addition to the material cost of the steel, it costs $2 per meter to weld these seams. Find the dimensions of the drum that will minimize the cost of production (materials and welding).
3.The cost of shipping each drum from your manufacturing plant to the oil company depends on both the surface area of the drum (which determines how much each drum weighs) and the sum of the diameter and height (which affects how many drums can be transported on one vehicle). This cost is $1 per square meter of surface area plus $0.50 per meter of diameter plus height. Find the dimensions of the drum that minimize the total cost of production and transportation.
This is what I have for the first question
A= 2r^2+2πrh
V= πr^2h -----> h=1/πr^2h
A=2r^2+2πr(1/(πr^2)-------> 2r^2+2/r
minimize 2r^2+(2/r)
A'=4r-2/r^2-----> LCD (4r^3-2)/(r^2)
critical points at A'=0
2(2r^3-1)=0 ------>
r=cuberoot(1/2)= (1/2)^1/3
h=1/π(1/2)^2/3
I'm stuck after that, please help!!
1.The cost of the steel used in making the drum is $3 per square meter. The top and bottom of the drum are cut from squares, and all the unused material from these squares is waste. The remainder of the drum is formed from a rectangular sheet of steel, assuming no waste. Ignoring all costs other than material cost, find the dimensions of the drum to minimizethe cost of materials(steel).
2.The drum has seams around the perimeter of the top and bottom, as well as a vertical seam where the edges of the rectangular sheet are joined to form the lateral surface. In addition to the material cost of the steel, it costs $2 per meter to weld these seams. Find the dimensions of the drum that will minimize the cost of production (materials and welding).
3.The cost of shipping each drum from your manufacturing plant to the oil company depends on both the surface area of the drum (which determines how much each drum weighs) and the sum of the diameter and height (which affects how many drums can be transported on one vehicle). This cost is $1 per square meter of surface area plus $0.50 per meter of diameter plus height. Find the dimensions of the drum that minimize the total cost of production and transportation.
This is what I have for the first question
A= 2r^2+2πrh
V= πr^2h -----> h=1/πr^2h
A=2r^2+2πr(1/(πr^2)-------> 2r^2+2/r
minimize 2r^2+(2/r)
A'=4r-2/r^2-----> LCD (4r^3-2)/(r^2)
critical points at A'=0
2(2r^3-1)=0 ------>
r=cuberoot(1/2)= (1/2)^1/3
h=1/π(1/2)^2/3
I'm stuck after that, please help!!