Optimizing Area in a Semi-Circle

In summary, the conversation is about finding 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. The suggested method is to use A=lw and to draw a diagram to see the mistake made. It is also mentioned that the problem can be solved using LaGrange multipliers.
  • #1
James99x
1
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:
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  • #2
Hint: start by drawing a diagram. You will see right away what you did wrong.

RGV
 
  • #3
Are you familiar with LaGrange multipliers? If so, then this problem is easier than it seems.
 
  • #4
Hi James99x! Welcome to PF! :smile:

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

Beyond that your method is fine! :wink:
 

FAQ: Optimizing Area in a Semi-Circle

What is an optimization word problem?

An optimization word problem is a type of math problem that involves finding the maximum or minimum value of a specific quantity given certain constraints or limitations.

What are some common examples of optimization word problems?

Examples of optimization word problems include finding the maximum profit for a business given certain production costs, determining the minimum amount of material needed to construct a box with a specific volume, and finding the shortest distance a car can travel given a certain amount of fuel.

What are the steps to solve an optimization word problem?

The general steps to solve an optimization word problem include identifying the objective function, writing down any constraints, finding the critical points of the objective function, determining which critical points are valid solutions, and checking the endpoints to ensure the maximum or minimum value is being achieved.

What are some strategies for solving optimization word problems?

Some strategies for solving optimization word problems include drawing a diagram or graph to visualize the problem, creating equations to represent the constraints and objective function, and using calculus techniques such as finding derivatives and solving equations to find the maximum or minimum value.

What are some common mistakes to avoid when solving optimization word problems?

Some common mistakes to avoid when solving optimization word problems include not carefully reading and understanding the problem, not setting up the equations correctly, and forgetting to check the endpoints for potential solutions.

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