Question
A cylinder shaped can needs to be constructed to hold 400 cubic centimeters of soup. The material for the sides of the can costs 0.03 cents per square centimeter. The material for the top and bottom of the can need to be thicker, and costs 0.06 cents per square centimeter. Find the dimensions for the can that will minimize production cost.
Answers
Reiny
let the radius be r
let the height be h
Volume = πr^2h
πr^2h = 400
h = 400/(πr^2)
cost = different prices x surface areas
= .03(2πrh) + 2(.06) πr^2
= .03[2πr(400/πr^2) + 4πr^2]
= .03[ 800/r + 4πr^2]
d(cost)/dr = .03[ -800/r^2 + 8πr] = 0 for a min of cost
800/r^2 = 8πr
100/π = r^3
r = 3.169
h = 400/(π(3.169)^2) = 12.679
check my arithmetic
let the height be h
Volume = πr^2h
πr^2h = 400
h = 400/(πr^2)
cost = different prices x surface areas
= .03(2πrh) + 2(.06) πr^2
= .03[2πr(400/πr^2) + 4πr^2]
= .03[ 800/r + 4πr^2]
d(cost)/dr = .03[ -800/r^2 + 8πr] = 0 for a min of cost
800/r^2 = 8πr
100/π = r^3
r = 3.169
h = 400/(π(3.169)^2) = 12.679
check my arithmetic
The Herald Of Cringe
this be bottted