Find the dimensions of the rectangular box with maximum volume in the first octant with one vertex at the origin and the opposite vertex on the ellipsoid 4x^2+y^2+4z^2=4 a. The length of the box in the x-direction is:
b. The length of the box in the y-direction is:
c. The length of the box in the z-direction is:

Solution :

Let x, y, z be the dimensions of the rectangle.

Volume of the rectangle (V) = xyz

Given that the vertex should lie on ellipse .......(i)

So here the volume xyz must be maximum with constraints above.

We solve this using Lagranches method with variable λ

Lagranches function is

.....(ii)

To find λ,

.............(iii)

..................(iv)

...............(v)

Equating (iii) and (iv)

...............(vi)

Equating (iii) and (v)

....................(vii)

Substitute (vi) and (vii) in (i),

From (i),

From (vi),

From (vii),

z = x

Therefore, for maximum volume the dimensions of a rectangle box are

answer: 1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

step-by-step explanation:

a)(x-2)^2+(y-1)^2=9

step-by-step explanation:

(picture explains)