/*! 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 21 Find the maxima: 1 possible area... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 21

Find the maxima: 1 possible area for a rectangle inscribed in the ellipse \(16 x^{2}+9 y^{2}=144\).

Short Answer

Expert verified
The maximum area of the rectangle is achieved at the critical point found in Step 5. Substituting this \(x\) value into the area function will yield the maximum possible area for the rectangle inscribed in the ellipse.

Step by step solution

01

Identify variables for the rectangle's length and width

Consider one side of the rectangle along the x-axis and the other side along the y-axis. In terms of a rectangle inscribed in the ellipse, the lengths of half of its sides (measured from the origin) are \(x\) and \(y\) respectively. Thus, the area of the rectangle can be represented by \(2xy\).
02

Express y in terms of x

Using the equation of the ellipse \(16x^{2} + 9y^{2} = 144\), express \(y\) in terms of \(x\) by rearranging the equation to: \(y = \sqrt{(144 - 16x^{2})/9}\).
03

Substitute y into the Area function

Substitute \(y\) from step 2 into the area function. You get a function of area in terms of \(x\), which is \(2x \cdot \sqrt{(144 - 16x^{2})/9}\).
04

Differentiate Area function

Differentiate the area function with respect to \(x\), this will give us a function to set equal to 0 in order to find the critical point that maximizes area. Using chain rule, the derivative of the area function, \(A'(x)\), is calculated as \(\frac{2 \cdot \sqrt{144 - 16x^{2}}}{9} - \frac{4x^{2} \cdot \sqrt{144 - 16x^{2}}}{(144 - 16x^{2})^{3/2}}\).
05

Find critical point

To find the maximum value, set the derivative of the area function, \(A'(x)\), equal to zero and solve for \(x\). This will be the x-coordinate of the rectangle vertex where maximum area occurs.
06

Validate critical point

Substitute the value of \(x\) obtained in step 5 into the area function to get the corresponding maximum rectangle area.

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.

Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This field forms the basis for much of modern mathematics and is applicable in various disciplines, including science and engineering.

Understanding calculus is vital for solving many real-world problems, including those involving rates of change and accumulation. For example, when considering the shape of an ellipse or the area of an inscribed rectangle, calculus helps us determine the maximum or minimum values of these geometric forms by employing differentiation and integration techniques.
Optimization Problems
Optimization problems are mathematical questions concerned with finding the best solution from a set of feasible alternatives. These problems often involve determining the maximum or minimum values of a function within a given domain.

In the context of calculus, optimization problems usually require finding local maxima or minima of a function, which correspond to the highest or lowest points in a particular range. The solution involves differentiating the function, determining the critical points, and evaluating these points to ascertain whether they are maxima or minima. The process of finding the largest possible area of a rectangle inscribed in an ellipse is an excellent example of an optimization problem.
Ellipse
An ellipse is a curve on a plane surrounding two focal points in such a way that the sum of the distances to the points (foci) from any point on the curve is constant. Ellipses are the closed type of conic sections, shapes that can be formed by intersecting a cone with a plane.

In the case of our optimization problem, the equation of an ellipse standard form is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a\) and \(b\) are the semi-major and semi-minor axes. The given equation \(16x^{2}+9y^{2}=144\) can be simplified to fit this form, which is essential for understanding how to inscribe a rectangle within it and further solve for its area.
Differentiation
Differentiation is the process of finding the derivative of a function. It measures how a function changes as its input changes. Loosely speaking, a derivative can be seen as how much the function 'wiggles' or 'slopes' at a certain point.

The derivative is pivotal in solving optimization problems because it can help find the function's extremes. In our inscribed rectangle problem, differentiation is used to determine the point where the area of the rectangle is maximized. This involves taking the derivative of the area function with respect to \(x\) and finding where this derivative equals zero, which signifies the potential locations of maxima or minima.

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

Use a CAS to determine where: (a) \(f^{\prime \prime}(x)=0\) (b) \(f^{\prime \prime}(x)>0\) (c) \(f^{\prime \prime}(x)<0\) $$f(x)=x^{11}-4 x^{9}+6 x^{7}-4 x^{5}+x^{3}$$

A tour boat heads out on a 100 -kilometre sight-seeing trip. Given that the fund costs of operating the boat total 2500 dollar per hour, that the cost of fuel varies directly with the square of the speed of the boat, and at 10 kilometres per hour the cost of the fuel is 400 dollar per hour, find the speed test minimizes the boat owner's expenses. Is the speed that minimizes the owner's expenses dependent on the length of the trip?

Two cars, car \(A\) traveling cast at \(30 \mathrm{mph}\) and car \(B\) traveling north at \(22.5 \mathrm{mph}\), are heading toward an intersection \(1 . \mathrm{At}\) what rate is the angle \(I A B\) changing when cars \(A\) and \(B\) arc 300 feet and 400 feet, respectively, from the intersection?

Set \(f(x)=\sin x+\frac{1}{2} x^{2}-2 x\). (a) Show that \(f\) has exactly one critical point \(c\) in the interval (2,3) (b) Use the Newton-Raphson method to estimate \(c\) by calculating \(x_{3} .\) Round off your answer to four decimal places. Does \(f\) have a local maximum at \(c,\) a local minimum, or neither?Does \(f\) have a local maximum at \(c, a\) local minimum, or neither?

(Simple harmonic motion) A bob suspended from a spring oscillates up and down about an equilibrium point, its vertical position at time \(t\) given by $$y(t)=A \sin \left(\omega t+\varphi_{0}\right)$$ where \(A, \omega, \varphi_{0}\) are positive constants. (This is an idealization in which we are disregarding friction.) (a) Show that at all times \(t\) the acceleration of the bob \(y^{\prime \prime}(l)\) is related to the position of the bob by the equation $$y^{\prime \prime}(t)+\omega^{2} y(t)=0$$. (b) It is clear that the bob oscillates from \(-A\) to \(A,\) and the speed of the bob is zero at these points. At what position does the bob attain maximum speed? What is this maximum speed? (c) What are the extreme values of the acceleration function? Where does the bob attain these extreme values?
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Applied 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.