Problema Solution
You have been hired to design a cylindrical tank to contain 3000 litres of oil. You want to minimize the amount of steel used. Find the dimensions of a cylindrical tank that will optimize surface area and volume (1000litres = 1m cubed)
Answer provided by our tutors
let
r = the radius of the cylinder
h = the height of the cylinder
V = 3000 l = 3 m^3 the volume of the cylinder
V = r^2 pi h
r^2 pi h = 3
h = 3/(r^2 pi)
A = 2r^2*pi + 2r pi*h the area of the cylinder
A = 2r^2*pi + 2r pi*(3/(r^2 pi))
A = 2r^2*pi + 6/r
lets find A' derivative of A
A' = 4 pi*r - 6/(r^2)
4 pi*r - 6/(r^2) = 0
4*pi*r^3 = 6
r^3 = 3/(2pi)
r = 0.7812 m
r = 78.12 cm
h = 3/(0.7812^2*3.14)
h = 1.5655 m
h = 156.55 cm
the dimensions of the cylinder tank are radius 78.12 cm and height 156.55 cm.