Question

Find the largest box (with faces parallel to the coordinate axes) that can be inscribed in $$\frac{x^{2}}{4}+\frac{y^{2}}{9}+\frac{z^{2}}{25}=1.$$

Short answer

The largest box dimensions are a=4\text{Sqrt}2, b=6\text{Sqrt}2, c=10\text{Sqrt}2 with volume = 720.

Step-by-step solution

- Understanding the Problem

The objective is to find the dimensions of the largest box that can be inscribed inside the ellipsoid given by \(\frac{x^{2}}{4}+\frac{y^{2}}{9}+\frac{z^{2}}{25}=1\). The faces of the box are parallel to the coordinate axes.

- Box Dimensions and Coordinates

Assume the box has its corners at points \((\frac{\text{a}}{2}, \frac{\text{b}}{2}, \frac{\text{c}}{2})\) and \((-\frac{\text{a}}{2}, -\frac{\text{b}}{2}, -\frac{\text{c}}{2})\). The center of the box is at the origin.

- Inscribed Condition

At any corner of the box, say \((\frac{\text{a}}{2}, \frac{\text{b}}{2}, \frac{\text{c}}{2})\), it must satisfy the ellipsoid equation. Thus: \(\frac{\left(\frac{\text{a}}{2}\right)^{2}}{4} + \frac{\left(\frac{\text{b}}{2}\right)^{2}}{9} + \frac{\left(\frac{\text{c}}{2}\right)^{2}}{25} = 1\).

- Simplifying the Equation

Expand and simplify the equation to obtain inequality: \( \frac{a^{2}}{16} + \frac{b^{2}}{36} + \frac{c^{2}}{100} = 1 \).

- Maximizing the Volume

The volume of the box is given by \(V = abc\). To maximize this volume under the given constraint, apply the method of Lagrange multipliers.

- Setting Up Lagrange Multipliers

Define the function \(f(a, b, c) = abc\) and the constraint \(g(a, b, c) = \frac{a^{2}}{16} + \frac{b^{2}}{36} + \frac{c^{2}}{100} - 1 = 0\). Use Lagrange multiplier \(\lambda\) such that \(abla f = \lambda abla g\).

- Solving the System of Equations

Calculate the partial derivatives: \( abla f = \left( bc, ac, ab \right) \) and \( abla g = \left( \frac{a}{8}, \frac{b}{18}, \frac{c}{50} \right)\). Equate: \( bc = \lambda \frac{a}{8} \), \( ac = \lambda \frac{b}{18} \) and \( ab = \lambda \frac{c}{50} \).

- Solving for Variables

Express \( \lambda \) from each equation and equate: \( \frac{bc8}{a} = \frac{ac18}{b} = \frac{ab50}{c} \). Solve the system to find: \( a = 2\text{Sqrt}{2} \), \( b = 3\text{Sqrt}{2} \) and \( c = 5\text{Sqrt}{2} \). Finally, dimensions are \( a = 4\text{Sqrt}{2} \), \( b = 6\text{Sqrt}{2} \) and \( c = 10\text{Sqrt}{2} \).

- Confirming Volume and Coordinates

The volume of the largest box is \( V = 4\text{Sqrt}{2} \cdot 6\text{Sqrt}{2} \× 10\text{Sqrt}{2} \) yielding a maximum volume subject to ellipsoid, volume of largest inscribed box = 720

Key concepts

Lagrange Multipliers

When you need to optimize a function (like maximizing or minimizing) subject to constraints, Lagrange multipliers are a powerful method. In this context, the function we want to maximize is the volume of the box, given by \(V = abc\). The constraint is from the ellipsoid equation, \( \frac{x^{2}}{4}+\frac{y^{2}}{9}+\frac{z^{2}}{25}=1 \).

Here's how Lagrange multipliers work:

• First, you express the function you wish to optimize as \(f(a, b, c)\).
• Next, write the constraint as \(g(a, b, c) = 0\).
• Then, introduce a new variable, \(λ\), called the Lagrange multiplier.
• Finally, solve the equation \( ∇ f = λ ∇ g \) which gives a system of equations to find the optimum point.

In our example, let \( f(a, b, c) = abc \) and \( g(a, b, c) = \frac{a^2}{16} + \frac{b^2}{36} + \frac{c^2}{100} - 1 = 0 \). Solving \( ∇ f = λ ∇ g \), we get the conditions necessary to find the dimensions of the box.

Ellipsoid

An ellipsoid is a three-dimensional geometric surface, similar to an elongated or flattened sphere. Its equation is of the form \( \frac{x^{2}}{A}+\frac{y^{2}}{B}+\frac{z^{2}}{C}=1 \).

Here, the given ellipsoid is \( \frac{x^{2}}{4}+\frac{y^{2}}{9}+\frac{z^{2}}{25}=1 \), which means:

• Along the x-axis, the lengths extend from \(-2\) to \(2\).
• Along the y-axis, the lengths extend from \(-3\) to \(3\).
• Along the z-axis, the lengths extend from \(-5\) to \(5\).

Understanding these stretches helps visualize the ellipsoid and how it can tightly fit the box within its structure. The center of the ellipsoid, like the box, is also at the origin (0,0,0).

Volume Maximization

Maximizing volume is vital when determining the largest possible box that can fit within the ellipsoid. The volume of a box is given by \(V = length \times width \times height\), or in terms of our box along the coordinate axes, \( V = abc \).

To find the maximum volume under the constraint \( \frac{a^2}{16} + \frac{b^2}{36} + \frac{c^2}{100} = 1 \), we use the Lagrange multipliers technique.

Setting up the system \( ∇ f = λ ∇ g \) yields three equations:

• \( bc = λ \frac{a}{8} \)
• \( ac = λ \frac{b}{18} \)
• \( ab = λ \frac{c}{50} \)

Solving this system for \(a\), \(b\), and \(c\) results in the optimal dimensions of the box.

Inscribed Box

An inscribed box is one that fits entirely within another shape, such as our ellipsoid. For this exercise, we want the largest box with faces parallel to the coordinate axes.

The box's vertices are \(\left(\frac{a}{2}, \frac{b}{2}, \frac{c}{2}\right)\) and \(\left(-\frac{a}{2}, -\frac{b}{2}, -\frac{c}{2}\right)\), and these must satisfy the ellipsoid equation.

Let's put this into steps:

• Express the constraint in the form \( \frac{x^{2}}{4}+\frac{y^{2}}{9}+\frac{z^{2}}{25}=1 \). This ensures the box's corners are inside the ellipsoid.
• Utilize Lagrange multipliers, where we set \( f(a, b, c) = abc \) and constrain \( g(a, b, c) \) using the ellipsoid equation.
• After solving, you find the box's dimensions maximize the volume while adhering to the constraint.

Conclusively, the dimensions \(a = 4\sqrt{2}\), \(b = 6\sqrt{2}\), and \(c = 10\sqrt{2}\) yield the largest possible volume for our inscribed box.