Answer :
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;
[tex]fx = \lambda gx[/tex]
where;
[tex]f: V = xyz[/tex]
[tex]g : x + 9y + 4z = 27[/tex]
From; [tex]fx = \lambda gx[/tex]
[tex]yz = \lambda[/tex] --------- equation (1)
From; [tex]fy = \lambda gy[/tex]
[tex]xz = 9 \lambda[/tex] --------- equation (2)
From; [tex]fz = \lambda gz[/tex]
[tex]xy = 4 \lambda[/tex] --------- equation (3)
Comparing and solving equation (1),(2) and (3);
[tex]\lambda x = 9 \lambda y = 4 \lambda z[/tex]
divide through by [tex]\lambda[/tex]
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