Mathematics 24x7

There is no escape...believe it or not. Mathematics is everywhere.

The two following problem permit to know how you work your way in maths.
Explanation came after the two problems


Triangle

Can you find wich configuration of the 3 points gives the greater area for the triangle A,B,C ?
Can you prove your it ?

Trapezoid


Given the following picture of a trapezoid:

What's the area of the greyed surface if she'is enclosed by segments joining middle of each side of trapezoid ?

Explanations

Don't read bellow before having answered both problems.

Circle problem

This problem will trap people used to "recognise and apply" approach.
When solving problem, you first attempt to classify it then use a predefined algorithm/set of tools.


This problem is classified as "Optimisation kind problem" thus you're going to use "Integral and algebra".
It's possible but very long and error prone :-)


The easiest way is to wonder if you can find some usable properties. Let's simplify the problem by fixing two points and moving the last one.
You will quickly see that the area is maximum when the third point is fare away of base (wich is fixed length since the two point are fixed). To be far away, the point must be on base bisector.


Since you can fix any two point of the 3, it should be true for any chosen couple ! The only triangle having his vertices on his sides bisector is the equilateral one.


Trapezoid problem

This one trap people using short-cuts :-)


The exhibited trapezoid does NOT exist


Answer is then simple : Area does not exist


Proving it is a little harder


Demo


First let's notice that g=h, I'll refer to g as h

|DA| = 5 and |BC| = 10

|DC| = 8

|AB| = |EF| = 4

Let's define |DE|=x then |FC|=4-x


Now some relations that use Pythagoras theorem :

25=h²+x²
100=h²+(4-x)²
Let's isolate h² so we can try to find a suitable "x" value.
h²=25-x²
h²=100-(16+x²-8x)
Search for "x" value :
25-x²=100-16-x²+8x
25=84+8x
So x is : -7.375
Let's replace that value in first identity :
h²=25-( -7.375)²
Not possible in R...





What you should keep in mind

Solving a problem once is not enough, solve it in different ways, with different approach.
If it's too easy, maybe you're missing something. Check and solve again.

Some problem have more than one solution, other doesn't have any. (Sound stupid but most expect the find The solution instead of A solutions)

Rewrite the problem with your own sentences and vocabulary, appropriate it.

Don't be afraid to drop your previous work if you feel that this is a dead end.
Don't be afraid to work on multiple approach at same time. It's seems odd at first glance but it's quicker and you can check your result afterwards by solving problem twice or more.

Sometime, take a break, look at new data you got wile solving and ask yourself if you can use them somewhere in you development.

Avoid routine.

Practice !


1) What's the area of the triangle having his vertices at the following coordinates : A(1,2); B(2,5); C(5,3) ?
2) The picture bellow represent a city map. I want to go from A to B. Each street I take must bring me toward B never toward A.(Only going East and South).
How many different way can I borrow ?


3) Create your own and post it ;-)

Remark :
I'm not a teacher neither a Mathematician. I simply love mathematics :-)

Views: 8

Comment by Colin McAllister on July 16, 2009 at 1:30am

Re: Triangle on circle, of which the problem is to identify the triangle of greatest area. One way of solving this is to use the physical properties of soap bubbles. This is a thought experiment, but it is plausible that it could be constructed by the art of glass blowing. It is well known from experiment that a soap bubble, i.e. a film of soapy water, will, due to surface tension, find its equilibrium in the position of smallest area. So, we must turn the problem inside out, and find the configuration that has the smallest area outside of the triangle. It is evident that the solution is at least isosceles, as any deviation of the apex will move it away from the tangent that is parallel to the opposite base, diminishing the height. Consider all possible isosceles triangles, from the vertical full diameter, to the diminishing horizontal arc, and those in between including the equilateral triangle (60, 60, 60) and right triangle (45, 90, 45). Cut a population of these triangles out of a thin sheet, and stack them end to end, building a contoured solid that, graduates through all the triangles, in sequence. Use a hollow tube to represent the circle, and enclose the triangular solid in the tube. The tube must be slightly larger than the circle, so that it does not contact the solid, permitting free movement of the soap bubble in the surrounding gap. The solid and cylinder may be considered welded, at one end, to a plane base that holds them together. Dip the open end of the cylinder into soapy water so that a plane bubble (a film) spans the gap between the cylinder and the enclosed solid. Ignore such forces as viscosity, gravity and air pressure, and assume that the film is always a plane parallel to the base of the cylinder. Surface tension will cause the film to move along the cylinder, to find its equilibrium position, the position of smallest area. In cross section, the film will be in the circle, in the area outside the triangle of greatest area. Thus we have used the physical properties of soap bubbles to solve the problem.
Comment by Colin McAllister on July 16, 2009 at 5:14am
Of course, the above solution using a film of soapy water is overkill. Once you have deduced that the triangle is isosceles, one more deductive step shows that the solution is equilateral. "Since you can fix any two point of the 3, it should be true for any chosen couple ! ", as Christian stated in his post. And, if you have gone to the trouble of cutting out all of the triangles from a sheet, you could simply weigh them to find the heaviest one.
There are other problems for which a soap bubble is an appropriate solution, for example:
Identify the 3-D shape which has the least surface area for a given volume.
Identify the 2-D shape which has the greatest area for a fixed circumference.
The mathematical properties of bubbles (Some results on Bubbles) have even been put to effective use in architecture, as in the "BEIJING BUBBLE BUILDING: China’s National Swim Center".

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