Question

if the length of the diagonal of a rectangular box must be l, use lagrange multipliers to find the largest possible volume.

Answer 1

To find the largest possible volume of a rectangular box with a given length of the diagonal, we can use Lagrange multipliers.

Step-by-step explanation:

To find the largest possible volume of a rectangular box with a given length of the diagonal, we can use Lagrange multipliers. Let the length, width, and height of the box be L, W, and H respectively. We want to maximize the volume V = LWH subject to the constraint L² + W² + H² = l² (since the length of the diagonal is l).

We can set up the Lagrange function as follows:

Λ(L, W, H, λ) = LWH + λ(L² + W² + H² - l²)

Then, we can take the partial derivatives of Λ with respect to L, W, H, and λ, and set them equal to zero to find the critical points. From there, we can determine which critical point gives the maximum volume.

Answer 2

To find the largest possible volume of a rectangular box with a given length of the diagonal, we can use Lagrange multipliers. We set up the Lagrangian function and take partial derivatives to find the values of the dimensions that maximize the volume.

Step-by-step explanation:

To find the largest possible volume, we can use the method of Lagrange multipliers. Let's define the length, width, and height of the rectangular box as L, W, and H respectively. The volume of the box is given by V = LWH. We need to maximize V under the constraint that the length of the diagonal is l. The constraint equation can be written as L² + W² + H² = l². We can set up the Lagrangian function as:

L = V + λ(L² + W² + H² - l²)

where λ is the Lagrange multiplier. Now, we can take partial derivatives of the Lagrangian with respect to L, W, H, and λ, set them equal to zero, and solve the resulting equations to find the values of L, W, and H that maximize the volume.