A cylindrical can is to be made to hold 1L of oil. Find the height of the can, in centimeters, that will minimize the cost of the metal to manufacture the can. Assume the base, sides and top are made of the uniformly thick metal and ignore seams. 1L=1000cm^3

To construct the container, a rectangle-shaped piece is cut to make the cylindrical body of said container, and two circular-shaped pieces for its lids. We establish A as the total area of the container, At will be the area of the lids, Ac will be the area of the body, r will be the radius of the cylindrical body and h will be its height.

The total area A of the cylindrical container will be given by the following equation:

A = 2 π r^2 + 2 π r h

To solve the equation we will use the information from the exercise: the volume of the container is 1 liter. This means that:

Volume = π r^2 h = 1000

Clearing the height h, we have:

h = 1000/(π r^2)

Substituting this expression in the equation for area A:

A = 2 π r^2 + 2 π r h = 2 π r^2 + 2 π r (1000 / (π r^2)) = 2 π r^2 + 2000/r

A(r) = 2 π r^2 + 2000/r, condition r> 0

Optimizing this equation:

A’(r) = 4 π r - 2000/r^2 = 4 (π r^3 - 500)/r^2

If A’(r) = 0

Solving for r:

r = 3√(500/π) cm

Substituting in the equation for height:

h = 1000/(π r^2),

h = 2 3√ (500/π) = 2 r cm