Tuesday, November 13, 2007

Drum Roll Please: The Debut of TC's Total Challenge

As you may have read in an earlier comment, I've invited one of MathNotations' most dedicated and talented contributors to go beyond commenting and share some of his creative ideas and insights by being an occasional guest blogger - he has graciously accepted.

For his inaugural offering, tc is challenging you and/or your students to solve a classic calculus problem using non-calculus methods. I have made a few minor edits, but the activity is essentially what tc sent to me.
I give you tc's Total Challenge I:

One of my math professors in college used to say there were three ways
of tackling any problem: the right way, the wrong way and the Navy way
(correct, but extremely roundabout).

In this exercise, we will look at three ways (not necessarily the ones
named above) of doing the following problem:

Determine the rectangle of maximum area that can be inscribed
in a circle of given radius r.

Let the inscribed rectangle have sides a and b. The diagonal of the rectangle passes through the center of the circle (this can be shown, but you can assume it is true).

(1) Express r in terms of a and b.

(2) Express the area in terms of a and r.

(3) Instead of maximizing the area, we can maximize the square of the area.
(a) Express the square of the area as a quadratic in a2 (you may want to substitute c for a2).
(b) By completing the square, determine the value of a for which the area is a maximum.
(c) Determine the value of b and the maximum area.
(d) What conclusion can you draw about the rectangle of maximum area?
(This is the first way, which I call the Algebra way)

(4) Divide the rectangle into 2 congruent triangles, using a diagonal. Draw a half
diagonal that intersects this diagonal.
(a) Write an inequality for the area of one of these triangles in terms of r alone. The inequality should be of the form Area ≤ _______.
(b) If you can achieve equality, then you have maximized the area of the rectangle! Find out when this occurs, and if it does, find the lengths of a and b. (The Geometry way).

(5) Method 3 - the Calculus way of course.
Additional comments from DM:
(i) thanks, tc!
(ii) tc's geometric approach in (4) also suggests a connection to the famous AM-GM Inequality. Visit this link and see if you can make the connection. This is not obvious.
Hint: Apply the AM-GM to a2 and b2.


Totally_Clueless said...

Hi Dave,

Thanks for the opportunity. Though I am not a math educator, I use lots of applied math in my work, and enjoy math for the sake of math. I hope your readers will also enjoy the problems I pose.

As a rule, and especially over the last week, I have found that formulating good problems is orders of magnitude more difficult than solving them. This gives me added appreciation for what you do on this blog, and I hope I can keep up with the standards you have set.


Dave Marain said...

Thanks, tc, for those kind and very generous words. Writing carefully worded, challenging yet appropriate investigations for middle and high school students is still arduous for me and I've been doing it for decades. On a smaller scale, educators know how difficult it is to ask the 'right' question at the 'right' time to the 'right' student, and when that fails, to be somehow able to rephrase the question in a way that will enable the student to 'see' it (or to encourage another student to rephrase the question!). My instructional methods course didn't quite prepare me for that.

However, for one who is not an educator by title, you are an educator in spirit. You love the subject and enjoy conveying its beauty. You're also very good at posing tough problems. I know how much I've struggled with some of them!

I hope our association will continue. I think it is beneficial to have more than one voice presenting challenges and activities here. I'm hoping we can make some Wednesdays Totally Clueless Activities Days! We can proceed slowly at first and see what kind of feedback we get. I suspect our readers will look forward to those Wednesdays!!

By the way, we could solicit ideas for the best name for your challenges. If anyone out there would like to offer a suggestion, we can make this into a contest and... Yes, I'm getting carried away here, but I'd really like to see some creative ideas for tc. Waiting...