Optimization: Rectangle Inscribed in Triangle

In summary: We know how to derive the quadratic equation from completing the square. It's tough going if you never learned it in school.
  • #1
phyzmatix
313
0
[SOLVED] Optimization: Rectangle Inscribed in Triangle

Homework Statement



Please see http://www.jstor.org/pss/2686484 link. The problem I have is pretty much exactly the same as that dealt with in this excerpt.

(focus on the bit with the heading "What is the biggest rectangle you can put inside a triangle")

Homework Equations



Shown in the link above.

The Attempt at a Solution



I basically want someone to please explain why we need not use a derivative. As you can see, the last sentence is chopped off and leaves me hanging. :smile:
 
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  • #2
We see that x(a-x) is maximum if and only if after completing the square [tex](x-\frac{a}{2})^2 =0[/tex] for when x=? Therefore, the maximum rectangle has a height of what?
 
  • #3
Sorry, I didn't really answer your question. You don't need to use the derivative to find the maximum values of x and y because by completing the squares you can find the maximum value of x. Then, you can use x to find the area of that maximum triangle in terms of the area of the triangle.
 
  • #4
Could you still use a derivative though?
 
  • #5
"We need not use a derivative" does not imply that we can't use a derivative to solve for maximum x. So, yes you can use a derivative.
 
  • #6
konthelion said:
"We need not use a derivative" does not imply that we can't use a derivative to solve for maximum x. So, yes you can use a derivative.

Thank you very much for your help konthelion. I'm going to give it a shot using derivatives (I know nothing about completing the squares) and if you don't mind, I'd like you to have a peek at it as soon as I get round to posting it here...
 
  • #7
It's very strange that a person would be able to use the derivative (a calculus topic) but not know how to complete the square (an algebra topic).
 
  • #8
HallsofIvy said:
It's very strange that a person would be able to use the derivative (a calculus topic) but not know how to complete the square (an algebra topic).

I know the quadratic equation, but never learned how it was derived (after some googling yesterday, I realized it's the result of completing the square)

Were never taught it in school. Also, there's an eight year gap between the last time I did any maths and starting my BSc in Physics this year...

It's tough going :smile:
 

Related to Optimization: Rectangle Inscribed in Triangle

1. How do you find the maximum area of a rectangle inscribed in a triangle?

To find the maximum area of a rectangle inscribed in a triangle, you need to follow these steps:

  • Find the base and height of the triangle.
  • Divide the base by 2 to get the length of the base of the rectangle.
  • Use the Pythagorean theorem to find the length of the rectangle's height.
  • Multiply the base and height to get the maximum area of the rectangle.

2. What is the formula for finding the area of a rectangle inscribed in a triangle?

The formula for finding the area of a rectangle inscribed in a triangle is A = 1/2 * b * √(a^2 - (b/2)^2), where A is the area of the rectangle, b is the base of the triangle, and a is the height of the triangle.

3. How do you prove that the rectangle inscribed in a triangle has the maximum area?

To prove that the rectangle inscribed in a triangle has the maximum area, you can use the following steps:

  • Assume that there exists a rectangle with a larger area inscribed in the same triangle.
  • Use the Pythagorean theorem to calculate the length of the rectangle's height.
  • Compare the area of this rectangle to the area of the original rectangle inscribed in the triangle.
  • Show that the area of the original rectangle is greater, thus proving that it has the maximum area.

4. Can a rectangle be inscribed in any type of triangle?

No, a rectangle cannot be inscribed in any type of triangle. It can only be inscribed in a right triangle or an isosceles triangle where the base is twice the height. This is because the rectangle's diagonal must be equal to one of the triangle's sides in order for it to be inscribed.

5. How is optimization related to finding a rectangle inscribed in a triangle?

Optimization is the process of finding the maximum or minimum value of a function. In the case of a rectangle inscribed in a triangle, the function is the area of the rectangle. By finding the maximum area of the rectangle inscribed in a triangle, we are optimizing the value of the function, thus connecting optimization to this problem.

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