Question

A cylindrical can has a volume of ___. Use Lagrange multipliers to determine the dimensions of the can that yield the minimum surface area?

Answer 1

To determine the dimensions of the cylindrical can that yield the minimum surface area, we can use Lagrange multipliers. This method involves setting up an equation with the surface area and volume of the can, taking partial derivatives, and solving for the critical points. By following these steps, we can find the dimensions of the can that minimize the surface area.

Step-by-step explanation:

To determine the dimensions of the cylindrical can that yield the minimum surface area, we need to use Lagrange multipliers.

1. First, we need to define the objective function, which is the surface area of the can. Let's call it S.
2. Next, we need to define the constraint function, which is the volume of the can. Let's call it V.
3. Use the Lagrange multiplier method to find the critical points of the objective function subject to the constraint. Set up the following equation: S - λ(V - k) = 0, where λ is the Lagrange multiplier and k is the given volume.
4. Take the partial derivatives of S, V, and the constraint function with respect to the variables (radius and height in this case).
5. Equate the partial derivatives to zero and solve the resulting system of equations to find the critical points.
6. Check the second partial derivatives to determine if the critical points will yield a minimum surface area.

By following these steps, you can find the dimensions of the cylindrical can that yield the minimum surface area.