Find the radius and height of a cylindrical soda can with a volume of 256cm^3 that minimize the surface area.

b: compare your answer in part a to a real soda can, which has a volume of 256cm^3, a radius of 2.8 cm, and a height of 10.7 cm, to conclude that real soda cans do not seem to have an optimal design. then use the fact that real soda cans have a double thickness in their top and bottom surfaces to find the radius and height that minimizes the surface area of a real can (the surface area of the top and bottom are now twice their values in part a.

b: new radius=?

new height=?

A) Radius: 3.44 cm.

Height: 6.88 cm.

B) Radius: 2.73 cm.

Height: 10.92 cm.

Step-by-step explanation:

We have to solve a optimization problem with constraints. The surface area has to be minimized, restrained to a fixed volumen.

a) We can express the volume of the soda can as:

This is the constraint.

The function we want to minimize is the surface, and it can be expressed as:

To solve this, we can express h in function of r:

And replace it in the surface equation

To optimize the function, we derive and equal to zero

The radius that minimizes the surface is r=3.44 cm.

The height is then

The height that minimizes the surface is h=6.88 cm.

b) The new equation for the real surface is:

We derive and equal to zero

The radius that minimizes the real surface is r=2.73 cm.

The height is then

The height that minimizes the real surface is h=10.92 cm.