Respuesta :
Answer:
The dimension of the given box is [tex]\sqrt{\frac{200}{3}}\ cm[/tex] by [tex]\sqrt{\frac{200}{3}}\ cm[/tex] by[tex]\sqrt{\frac{200}{3}}\ cm[/tex].
Step-by-step explanation:
Assume the dimension of the box is x by y by z.
Total surface area of the box= 2xy+2yz+2xz=400
The volume of the box is f(x,y,z)= xyz
To maximum volume under constrain,
g(x,y,z)=2xy+2yz+2xz-400=0
we use Langrange Multiplier.
Assume that,
[tex]\triangledown f=\lambda \triangledown g[/tex]
[tex]\Rightarrow <f_x,f_y,f_z>=\lambda <g_x,g_y,g_z>[/tex]
[tex]\Rightarrow <xy,yz,xz>=\lambda<2(y+z),2(x+z),2(x+y)>[/tex]
[tex]\therefore xy=2 \lambda (y+z)[/tex] [tex]\Rightarrow xyz= 2\lambda(xy+xz)[/tex]
[tex]\therefore yz=2 \lambda (x+z)[/tex] [tex]\Rightarrow xyz= 2\lambda (xy+yz)[/tex]
[tex]\therefore xy=2 \lambda (x+y)[/tex] [tex]\Rightarrow xyz= 2\lambda (xz+yz)[/tex]
[tex]\therefore xyz= 2\lambda (xy+xz)=2 \lambda (xy+yz)=2 \lambda (xz+yz)[/tex]
If [tex]\lambda \neq 0[/tex],
[tex]\Rightarrow (xy+xz)= (xy+yz)=(xz+yz)[/tex]
This implies that, x=y=z
Then 2xy+2yz+2xz=400
⇒xy+yz+xz=200
⇒x²+x²+x²=200
⇒3x²=200
[tex]\Rightarrow x=\sqrt {\frac{200}{3}}[/tex]
Therefore [tex]x=y=z=\sqrt{\frac{200}{3}}[/tex] cm.
