Answer:
36 x 18 x 18
Step-by-step explanation:
By Lagrange multiplier's method,
∇f(x,y,z)= λ∇g(x,y,z) & g(x,y,z)=k
considering x,y,z as three unequal sides of the box
f(x,y,z)=xyz
If x represents the length, then constraint condition is g(x,y,z)= x + 2(y+z) =108
We have the following three equations:
yz= λ -->eq(1) fx= λgx
xz=2λ -->eq(2) fy=λgy
xy= 2λ -->eq(3) fz=λgz
x+2y+2z=108-->eq(4)
Dividing eq(2) by eq(1), we'll have
x / y =2
=> x= 2y
Also, by using eq(2)and eq(3), we can represent y=z
By substituting 'x= 2y' and 'y=z' in eq(4), we have
eq(4)=>
2y+ 2(y+y)= 108
y=108/6
y=18
<u>For x: </u> x=2y=> 2(18)=>36
<u>For z: </u>z= y = 18
Therefore, the dimensions are 36 x 18 x 18
