Optimizing Area in a Semi-Circle

Click For Summary

Homework Help Overview

The problem involves finding the dimensions of a rectangle with the largest area that can be inscribed in the upper semi-circle defined by the equation x^2+y^2 ≤ 16, y≥0. The subject area pertains to optimization in geometry.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to use the area formula A=lw and expresses uncertainty about their approach. Some participants suggest drawing a diagram to clarify the problem, while others mention the use of LaGrange multipliers as a potential method. There is also a note about the original poster having the wrong base for the rectangle.

Discussion Status

The discussion is ongoing, with participants providing hints and suggestions for clarification. There are indications of multiple interpretations regarding the setup of the problem, and some guidance has been offered regarding the area calculation.

Contextual Notes

Participants are navigating assumptions about the dimensions and relationships within the semi-circle, and there is a mention of potential errors in the original poster's setup.

James99x
Messages
1
Reaction score
0
1. Find the dimensions of the rectangle with the largest area that can be inscribed in the upper semi-circle given by x^2+y^2 ≤ 16, y≥0.
2. I thought I'd use A=lw
3. This is but a guess..so take it with a grain of salt..

height=2x
base= x^2+y^2

A(x) = 2x(x^2+y^2)
= 2x^3+2xy^2

A'(x) = 6x^2+2y^2+4x(dy/dx)(y)

I'm not sure what to really do about this particular problem.
 
Last edited:
Physics news on Phys.org
Hint: start by drawing a diagram. You will see right away what you did wrong.

RGV
 
Are you familiar with LaGrange multipliers? If so, then this problem is easier than it seems.
 
Hi James99x! Welcome to PF! :smile:

You have the wrong "base" for your square...

Beyond that your method is fine! :wink:
 

Similar threads

Find the dimensions that will minimize the surface area of a Rectangle
  • · Replies 1 ·
Replies
1
Views
2K
Computing line integral using Stokes' theorem
  • · Replies 8 ·
Replies
8
Views
2K
Verify Green's Theorem in the given problem
  • · Replies 10 ·
Replies
10
Views
2K
Solve this problem that involves implicit differentiation
  • · Replies 15 ·
Replies
15
Views
2K
Variation of parameter VS Undetermined Coefficients
  • · Replies 7 ·
Replies
7
Views
2K
Integration problem using inscribed rectangles
  • · Replies 14 ·
Replies
14
Views
2K
Question on Two Differentiation Techniques
  • · Replies 25 ·
Replies
25
Views
2K
Solve the problem that involves partial fractions
  • · Replies 4 ·
Replies
4
Views
2K
Can't get Lagrange multiplier to work in a single exercise
  • · Replies 16 ·
Replies
16
Views
3K
Calculating Area Under a Curve Using Change of Variables and Quotient Rule
  • · Replies 5 ·
Replies
5
Views
2K