Use Lagrange multipliers to find the dimensions of a rectangular box with largest volume if the total surface area is given as 400 cm2. (Enter the dimensions (in centimeters) as a comma separated list.)

The dimension of the given box is by by.

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,

If ,

This implies that, x=y=z

Then 2xy+2yz+2xz=400

⇒xy+yz+xz=200

⇒x²+x²+x²=200

⇒3x²=200

Therefore cm.

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