Solve the problem.
-A rectangular box is to be inscribed inside the ellipsoid $2 x ^ { 2 } + y ^ { 2 } + 4 z ^ { 2 } = 12$. Find the largest possible volume for the box.
A) $16 \sqrt { 2 }$
B) $15 \sqrt { 2 }$
C) $12 \sqrt { 2 }$
D) $18 \sqrt { 2 }$

Question

Quizplus · Accepted Answer

First, note that the ellipsoid is symmetrical about each axis, so we only need to consider one octant of it. Also, note that the box must also be symmetrical about each axis, so the height, width, and length are each twice the same variable, say $a$. Then, the volume of the box is $V = 8a^3$.

To maximize this volume, we must find the maximum value of $a$ subject to the constraint that the box is inside the ellipsoid. One way to do this is to use Lagrange multipliers.

Consider the function $f(x,y,z) = 8a^3$ subject to the constraint $g(x,y,z) = 2x^2 + y^2 + 4z^2 - 12 = 0$. We want to find the maximum of $f$ subject to this constraint, which means we need to find the critical points of the function $L(x,y,z,\lambda) = f(x,y,z) - \lambda g(x,y,z)$.

Taking gradients, we get

$$\begin{aligned} \nabla f &= \langle 24a^2, 0, 0 \rangle \ \nabla g &= \langle 4x, 2y, 8z \rangle \ \end{aligned}$$

And so we need to solve the system of equations

$$\begin{aligned} 24a^2 &= 4\lambda x \ 0 &= 2\lambda y \ 0 &= 8\lambda z \ 2x^2 + y^2 + 4z^2 &= 12 \end{aligned}$$

If $\lambda = 0$, then we can immediately see that we must have $y = 0$ and $z = 0$ in order for the ellipsoid constraint to be satisfied. Moreover, since the ellipsoid is positive-definite, this point is a minimum of the function, not a maximum.

Otherwise, $\lambda \neq 0$ and so we must have $y = z = 0$ and $x^2 = 3/\lambda$, so

$$g(x,0,0) = 2x^2 = 6/\lambda.$$

Since $g$ must be 0, we have $\lambda = 2\sqrt 2$ and $x = \sqrt 6/\sqrt{2\sqrt 2}$, and hence

$$a = \frac {\sqrt 6}{2\sqrt{2\sqrt 2}}.$$

Then, the maximum volume is

$$V = 8a^3 = 16\sqrt 2.$$

So the answer is A).

Solve the Problem 2x2+y2+4z2=122 x ^ { 2 } + y ^ { 2 } + 4 z ^ { 2 } = 122x2+y2+4z2=12

Solve the Problem $2 x ^ { 2 } + y ^ { 2 } + 4 z ^ { 2 } = 12$