Applications of Method of Exhaustion

[number0]

Homework Statement

Show that the area of two triangles with the same base and height can be
approximated arbitrarily closely by the same set of rectangles, differently
stacked.

See attachment for picture!

Homework Equations

Use method of exhaustion (NOT proof by exhaustion).

The Attempt at a Solution

The book I have does not clearly explain what the method of exhaustion is. I looked online I still cannot find out how it is used except for the famous case where Archimedes used it to estimate the area of the circle or something related to that manner.

What I did was I ASSUMED that you can use a set of rectangles, call it M, to approximate the area of Triangle A.

Let N be define as the set of rectangles from M, but differently stacked.

Let Triangle B be define as a Triangle that has the same base and height as Triangle A.

Want to show that: Area(N) approximate Area(Triangle B).

By assumption, Area(M) approximates Area(Triangle A)

But, Area(Triangle A) = Area (Triangle B)

=> Area(M) approximates Area(Triangle B)

But, Area(M) = Area(N)

=> Area(N) approximates Area(Triangle B)

End.Since this problem is listed under the Method of Exhaustion section, I am certain that I did not use Method of Exhaustion and thus I am pretty sure I did something wrong or that I misinterpret the question or made a bold assumption or etc.

Anyone have an advice on how to implement the Method of Exhaustion for this particular problem or for any other problem in general? Thanks.

* Side note: From what I gathered so far, Method of Exhaustion assumes double contradiction to prove the equality. For instance, show that A < C is false and that C > A is false so that A = C is true. It has been noted that Method of Exhaustion is "like" taking the limit as n->infinity and is considered to be the precursor to calculus. I also have another homework problem that is asking for Method of Exhaustion but it is not related to this one.

Any advice would be appreciated! Thanks for your time and response.

 

[mighty2000]

Thank you for bringing this problem to my attention. It seems like you have a good understanding of the concept of Method of Exhaustion, but your attempt at a solution is not quite correct. Let me guide you through the correct approach for this problem.

Firstly, let's define the problem clearly. We want to show that the area of two triangles with the same base and height can be approximated arbitrarily closely by the same set of rectangles, differently stacked. In other words, we want to show that the area of Triangle A can be approximated by a set of rectangles, and that the same set of rectangles can also approximate the area of Triangle B.

Now, let's consider a set of rectangles, call it M, that can approximate the area of Triangle A. Since Triangle A has the same base and height as Triangle B, we can also use the same set of rectangles, M, to approximate the area of Triangle B. However, we want to show that we can also use a different set of rectangles, call it N, to approximate the area of Triangle B. This is where the Method of Exhaustion comes in.

We will use the method of exhaustion by contradiction to prove that N can approximate the area of Triangle B. Suppose that N cannot approximate the area of Triangle B. This means that there exists a non-zero difference between the area of Triangle B and the area of N. Let's call this difference ε.

Since we know that M can approximate the area of Triangle B, we can use M to approximate Triangle B to within an error of ε/2. This means that the area of Triangle B will be bounded by the area of M plus ε/2 and the area of M minus ε/2. In other words, we have:

Area(M) - ε/2 < Area(Triangle B) < Area(M) + ε/2

But we assumed that there exists a non-zero difference between the area of Triangle B and the area of N. This means that there exists a value of n (the number of rectangles in N) such that the area of N is larger than the area of Triangle B by a value greater than ε/2. In other words, we have:

Area(Triangle B) + ε/2 < Area(N)

But we also know that M can approximate the area of Triangle B to within an error of ε/2. This means that the area of N will also be bounded by the area