Question

Using the Lagrange Multiplier optimization method, design a

cylindrical can with the minimum surface area give the volume V
=r2h by expressing the height, h, in terms of the
radius, r.

Answer 1

To design a cylindrical can with the minimum surface area given the volume V = r^2h using the Lagrange Multiplier optimization method, we need to express the height, h, in terms of the radius, r. The process involves setting up the Lagrangian function, taking partial derivatives, solving for the variables, and substituting the result to find the minimum surface area.

Step-by-step explanation:

To design a cylindrical can with the minimum surface area given the volume V = r2h using the Lagrange Multiplier optimization method, we need to express the height, h, in terms of the radius, r.

Let's denote the surface area of the cylinder as S. The objective is to minimize S subject to the constraint V = r2h.

1. Define the Lagrangian function L as L = S + λ(V - r2h), where λ is the Lagrange multiplier.
2. Take the partial derivatives of L with respect to S, r, h, and λ.
3. Set the partial derivatives equal to zero and solve the resulting equations to find the values of r, h, and λ.
4. From the equation V = r2h, solve for h in terms of r (h = V/r2).
5. Substitute the value of h into the equation for S to find the minimum surface area.