Question

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.

Answer 1

Answer:

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;

$fx = \lambda gx$

where;

$f: V = xyz$

$g : x + 9y + 4z = 27$

From; $fx = \lambda gx$

$yz = \lambda$    --------- equation (1)

From; $fy = \lambda gy$

$xz = 9 \lambda$  --------- equation (2)

From; $fz = \lambda gz$

$xy = 4 \lambda$ --------- equation (3)

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

$\lambda x = 9 \lambda y = 4 \lambda z$

divide through by $\lambda$

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