Use Lagrange multipliers to find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the planex + 9y + 4z = 27.

81/4

Step-by-step explanation:

From the given information; we are to use Lagrange multipliers to find the volume of the largest rectangular box

The coordinate planes and the vertex given in the plane is x + 9y + 4z = 27.

By applying Lagrange multipliers, we have;

where;

From;

   --------- equation (1)

From;

 --------- equation (2)

From;

--------- equation (3)

Comparing and solving equation (1),(2) and (3);

divide through by

x = 9 y = 4z

3x = 27

x = 27/3

x = 9

From x = 9y

9 = 9 y

y = 9/9

y = 1

From

x = 4z

9 = 4 z

z = 9/4

Thus; the Volume of the largest rectangular box = 9 × 1  × 9/4

= 81/4

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