A rectangular box is to be inscribed inside the ellipsoid #2x^2 +y^2+4z^2 = 12#. How do you find the largest possible volume for the box?

Answer 1

#V = 16 sqrt(2)#

The box volume is given by

#V = 8 abs(x y z)#

so the problem is:

Find #max V(x,y,z)# subjected to
#g(x,y,z) = 2x^2 +y^2+4z^2 = 12#

Using Lagrange multipliers we have the equivalent problem

Find the stationary points of

#L(x,y,z,lambda) = V(x,y,z) + lambda g(x,y,z)#
and verify the solutions which give a maximum for #V(x,y,z)#
The stationary ponts are obtained by solving for #x,y,z,lambda#
#grad L(x,y,z,lambda) = vec 0# or

#{ (8 y z-4 lambda x =0), ( 8 x z-2 lambda y =0), (8 x y - 8 lambda z=0), (12 - 2 x^2 - y^2 - 4 z^2=0):}#

The solution is # (x = sqrt[2], y = 2, z = 1)# with corresponding volume

#V = 16 sqrt(2)#
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Answer 2

To find the largest possible volume for the box inscribed inside the ellipsoid (2x^2 + y^2 + 4z^2 = 12), we need to maximize the volume of the box subject to the constraint that it fits inside the ellipsoid.

The volume of a rectangular box is given by (V = l \times w \times h), where (l), (w), and (h) are the length, width, and height of the box respectively.

To inscribe the box inside the ellipsoid, we need to find the points on the ellipsoid that are farthest from the origin in each direction (i.e., along the (x), (y), and (z) axes), as these points will determine the dimensions of the box.

To find these points, we set each of the coordinates (x), (y), and (z) equal to zero, one at a time, and solve for the remaining variables. This gives us the points ((\pm\sqrt{6}, 0, 0)), ((0, \pm\sqrt{12}, 0)), and ((0, 0, \pm\sqrt{3})), which represent the vertices of the box.

The lengths of the sides of the box are then given by twice the absolute values of these coordinates: (2\sqrt{6}), (2\sqrt{12}), and (2\sqrt{3}).

Therefore, the largest possible volume for the box is (V = (2\sqrt{6}) \times (2\sqrt{12}) \times (2\sqrt{3}) = 48\sqrt{2}).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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