A manufacturer needs to produce a cylindrical can with a volume (capacity) of 1000cm cubed. The top and the bottom of the container are made of material that costs $0.05 per square cm, while the side of the container is made of material costing $0.03 per square centimeter. Find the dimensions of the container that will minimize the company's cost of producing this container. What is the minimum cost?
4 answers
minimum area
the problem is asking me to use optimization to find the minimum surface are and then determine how much it would cost to make it
let the radius of the cylinder be r,
let the height of the cylinger be h
πr^2h = 1000
h = 1000/(πr^2)
Cost = .05(2πr^2) + .03(2πrh)
= .1πr^2 + .06πr(1000/(πr^2)
= .1 πr^2 + 60/r
d(Cost)/dr = .2πr - 60/r^2
= 0 for a min cost
.2πr = 60/r^2
r^3 = 60/(.2π) = 300/π
carry on, find r, then sub into the simplified expression for Cost above
let the height of the cylinger be h
πr^2h = 1000
h = 1000/(πr^2)
Cost = .05(2πr^2) + .03(2πrh)
= .1πr^2 + .06πr(1000/(πr^2)
= .1 πr^2 + 60/r
d(Cost)/dr = .2πr - 60/r^2
= 0 for a min cost
.2πr = 60/r^2
r^3 = 60/(.2π) = 300/π
carry on, find r, then sub into the simplified expression for Cost above
How do I apply using derivative
1 answer