So Volume of cylinder is pir^2h = 1,000
Then lightest one means you have the smallest surface area. Which is one base and then the area of the surface. SA = pir^2 + 2pirh
So now you have 2 equations, so:
h = 1,000/(pir^2)
So then SA = pir^2 + 2pir(1,000/(pir^2) = pir^2 + 2,000/r
Derivative of SA is then 2pir -2,000/r^2. Set to 0
2pir-2,000/r^2 =0 --> 2pir^3 = 2,000 --> r^3 = 1,000/pi --> r = 10/pi^(1/3)
Now go back to the volume function: pir^2h =1,000 --> 1,000/(pi*100/pi^(2/3)) = h
h = 10 / pi^(1/3)