/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 11 The volume of an ellipsoid \(\fr... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 11

The volume of an ellipsoid \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1\) is \(4 \pi a b c / 3\). For a fixed sum \(a+b+c\), show that the ellipsoid of maximum volume is a sphere.

Short Answer

Expert verified
For a fixed sum \(a+b+c\), the ellipsoid of maximum volume is indeed a sphere with radius \(\frac{k}{3}\), where \(k\) is the fixed sum.

Step by step solution

01

Define the Objective Function

Firstly, let's define the volume \(V\) of the ellipsoid \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1 \) as \(V= \frac{4}{3}\pi a b c\). This is our objective function. It's the function we want to optimize.
02

Define the Constraint

Next, let's define the constraint. The problem states that the sum of the semi-axes is constant, that is, \(a+b+c=k\), where \(k\) is a constant. This is our constraint.
03

Use Lagrange Multipliers

To find the maximum of the volume under the constraint, we can use the method of Lagrange multipliers. This involves forming a new function by adding the constraint to the volume, multiplied by a new variable called the Lagrange multiplier (\(\lambda\)). The new function to be maximized (or minimized) is: \(L(a, b, c, \lambda) = \frac{4}{3}\pi a b c + \lambda (k - a - b - c)\).
04

Compute the Partial Derivatives

Next, we compute the partial derivatives of \(L\) with respect to \(a\), \(b\), \(c\), and \(\lambda\), and set each of them equal to zero (extrema occur where the derivative is zero). We thus obtain a system of four equations to solve.
05

Solve the System

Solving the system of equations, we find that the solution (under our constraints) is \(a=b=c=\frac{k}{3}\), indicating the ellipsoid becomes a sphere, where the radius is \(\frac{a+b+c}{3}\).
06

Verify the Maximum Volume

In order to ensure we have achieved a maximum, we must check the second derivatives, according to the second derivative test. However, since we have a sphere when \(a=b=c\), and knowing that a sphere has the minimum surface area for a given volume, it follows geometrically that the volume is at a maximum when the ellipsoid is a sphere. Hence, the ellipsoid attains its maximum volume when it is a sphere.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ellipsoid Volume
An ellipsoid is a three-dimensional surface where all cross-sections are ellipses or circles. When it comes to its volume, it's important to remember that it's similar to that of a sphere but needs adjustments based on its axis lengths.
The standard formula for calculating the volume of an ellipsoid given its semi-axes lengths, denoted as \(a\), \(b\), and \(c\), is:
  • \( V = \frac{4}{3} \pi a b c \)
This formula gives a good approximation of the space contained within an ellipsoid.
This specific exercise aims to investigate what happens if, for a given constant sum of the semi-axes \(a + b + c = k\), the volume is optimized. The realization that maximum volume is achieved when the ellipsoid is actually a sphere, \(a = b = c\), is a fascinating result that aligns well with the intuition about spheres occupying the maximum volume for the least surface area.
Optimization Problem
An optimization problem involves finding the best solution from a set of alternatives. In mathematics, and especially in calculus, we frequently optimize by maximizing or minimizing functions subject to certain constraints.
In this problem, our goal is to maximize the volume of an ellipsoid while adhering to the condition that the sum of its axes is constant, \(a+b+c=k\). The objective function, which we aim to optimize, is quite simple:
  • The volume formula: \( V = \frac{4}{3} \pi a b c \)
For students grappling with such problems, it's crucial to understand the relationship between the main function (or objective function) and the constraints. This relationship often drives how we frame and solve the problem, especially when using tools like Lagrange multipliers.
Constraint Optimization
Constraint optimization is a critical concept in mathematics, particularly in scenarios where we need to find the best solution under specific conditions or restrictions.
In this exercise, the constraint is the constant sum of the semi-axes lengths of the ellipsoid, \(a+b+c=k\). To solve the optimization problem with this constraint, we utilize the method of Lagrange multipliers.
This approach requires:
  • Creating a Lagrangian function that incorporates both the objective function and the constraint.
  • Setting the partial derivatives of this function with respect to all variables to zero.
  • Solving the resulting system of equations to find the values of \(a, b, \) and \(c\), which in this case turned out to be equal.
This method is particularly effective because it reduces a constrained problem to a series of derivatives, which is often more straightforward to manage. In our specific problem, it elegantly proves that a sphere鈥攚ith equal semi-axes鈥攊s the optimal solution for maximum volume under the given constraint.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The table shows the world populations \(y\) (in billions) for five different years. (Source: U.S. Bureau of the Census, International Data Base) $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { Year } & 1994 & 1996 & 1998 & 2000 & 2002 \\ \hline \text { Population, } \boldsymbol{y} & 5.6 & 5.8 & 5.9 & 6.1 & 6.2 \\ \hline \end{array} $$ Let \(x=4\) represent the year 1994 . (a) Use the regression capabilities of a graphing utility to find the least squares regression line for the data. (b) Use the regression capabilities of a graphing utility to find the least squares regression quadratic for the data. (c) Use a graphing utility to plot the data and graph the models. (d) Use both models to forecast the world population for the year \(2010 .\) How do the two models differ as you extrapolate into the future?

Use Lagrange multipliers to find the minimum distance from the curve or surface to the indicated point. [Hint: In Exercise 23, minimize \(f(x, y)=x^{2}+y^{2}\) subject to the constraint \(2 x+3 y=-1 .]\) $$ \begin{array}{ll} \underline{\text { Curve }} & \underline{\text {Point}} \\ \text { Circle: }(x-4)^{2}+y^{2}=4 \quad (0,10) \end{array} $$

Let \(T(x, y, z)=100+x^{2}+y^{2}\) represent the temperature at each point on the sphere \(x^{2}+y^{2}+z^{2}=50 .\) Find the maximum temperature on the curve formed by the intersection of the sphere and the plane \(x-z=0\)

Find the absolute extrema of the function over the region \(R .\) (In each case, \(R\) contains the boundaries.) Use a computer algebra system to confirm your results. \(f(x, y)=x^{2}+2 x y+y^{2}, \quad R=\left\\{(x, y): x^{2}+y^{2} \leq 8\right\\}\)

Use Lagrange multipliers to find the dimensions of a rectangular box of maximum volume that can be inscribed (with edges parallel to the coordinate axes) in the ellipsoid \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1\)
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.