P.o.t.D. - 150 Problems !
Sunday 30 July 2017
Over the past six months, I’ve toiled away at composing problems to add to my two Problem of the Day lists – one list of problems for Maths SL and a second list of problems for Maths HL. Last week I managed to produce my 150th Problem of the Day – and to mark that milestone I thought I would try and create a problem that was a true problem for myself. That is, a question for which I did not know the answer and not immediately aware of an effective solution strategy.
The problem that I devised was motivated by a problem that I found in a short article entitled How Do You Slice the Bread? The authors of the article model a slice of bread in two dimensions with a rectangle and a semi-ellipse (as shown in diagram at right). One of the questions they pose – and solve – is where to cut a slice of bread to produce two “triangular” pieces that are equal in area with the ‘cut’ starting from the lower left corner of the rectangular section.
This made me come up of with a similar question: Where do you cut a semicircle – also starting from the lower left ‘corner’ – to create two regions of equal area? Although someone may have posed and solved this problem previously, I’m not aware of such and was surprised not to find anything after searching on the internet. My work on solving the problem required me to put on my problem-solving hat and led to some engaging mathematics and a solution with an interesting and unexpected property. I became so engrossed with the problem that I ended up solving it two different ways.
I have posted my solutions and notes for this problem at: Semicircle Chord Cut v2 - Solutions & Notes. Check out my Geogebra applet, semicircle chord cut v2 (applet), which dynamically reveals the solution and illustrates the unexpected property of point P. [Note: This is v2 (version 2) because there is another semicircle dissection problem on the site, semicircle chord cut v1, where one needs to find a chord horizontal to the diameter that cuts the semicircle into two regions of equal area]