Optimization of Area; Inscribed Rectangles

In summary, the conversation discusses the solution to a geometry problem involving finding the maximum area of an inscribed rectangle within a semicircle. The method involves substituting the semicircle equation into the area equation and then differentiating with respect to x to find the optimal point for maximum area. The final solution is that the dimensions of the rectangle must be 2sqrt6 and sqrt6.
  • #1
Qube
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Homework Statement



http://i.minus.com/jbkHm5oH1LfQ1k.png

Homework Equations



The area of a rectangle is its base times its height.

The Attempt at a Solution



The rectangle is inscribed. Its area is 2xy. I can substitute in the equation of the semicircle to get rid of the y-term in my area equation. I can then differentiate with respect to x and find what zeros the derivative (or causes the derivative not to exist) and test the points I find that are on the domain of x (0 to square root of 12).

I find x = sqrt6 as that point that maximizes area and therefore the dimensions must be 2sqrt6 and sqrt6.

http://i.minus.com/jFWGaWoQsb5FX.jpg
 
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  • #2
Looks right to me.
 

Related to Optimization of Area; Inscribed Rectangles

1. What is the purpose of optimizing the area of an inscribed rectangle?

The purpose of optimizing the area of an inscribed rectangle is to find the maximum possible area within a given shape or boundary. This is useful in many real-world applications, such as designing efficient layouts for buildings or maximizing crop yield in farming.

2. What factors affect the optimization of an inscribed rectangle's area?

The two main factors that affect the optimization of an inscribed rectangle's area are the dimensions of the original shape or boundary and the proportions of the inscribed rectangle. Other factors may include any constraints or limitations given in the problem, such as a maximum perimeter or minimum area requirement.

3. How do you determine the dimensions of an optimized inscribed rectangle?

To determine the dimensions of an optimized inscribed rectangle, you can use the method of differential calculus. This involves finding the critical points of the area function, which are the points where the function's derivative is equal to zero. The dimensions can then be calculated using these critical points.

4. What is the difference between an inscribed rectangle and a circumscribed rectangle?

An inscribed rectangle is one that is drawn inside a given shape or boundary, while a circumscribed rectangle is drawn outside the shape or boundary. In optimization problems, we are typically concerned with inscribed rectangles because they provide the maximum possible area within the given constraints.

5. Can the optimization of an inscribed rectangle's area be applied to other shapes besides rectangles?

Yes, the concept of optimizing the area of an inscribed shape can be applied to other shapes besides rectangles. This includes circles, triangles, and other polygons. The method for finding the optimal area may vary depending on the shape, but the general principles of finding critical points and using calculus still apply.

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