Investigate the extremes

 

     One of the unconventional problem solving strategies is to investigate the extremes of the problem. This method requires a rather unusual way of thinking. You are going to modify variables that do not affect the problem’s solution but make the problem considerably simpler.

 

 

Question 1

 

             The tangent AB of the smaller of the two concentric circles is a chord of the larger circle. Find the area of the region in yellow, if AB = 4.

 

Traditional solution

 

Let the radius of the big circle be R and the radius of the small circle be r.

 

Since AB is a tangent,  OT ^ AB.

AB = 4, therefore AT = 2.

By Pythagoras Theorem, OA2 = OT2 + AT2

\        R2 = r2 + 22

\         R2 – r2 = 4              (1)

Area of the region in yellow    = pR2 - pr2

                                                     = p( R2 – r2)

                                                     = 4p, by (1)

 

 

 

 

 

“Investigate the extreme” method

 

             Assume the smaller circle is made smaller and smaller, and finally reduced to a point. This can be done without loss of generality, and the problem remains intact. Then AB becomes the diameter of the large circle.

             \ Area of the yellow region = area of the larger circle = p(2)2 = 4p

 

 

Question 2

 

Here is a more difficult problem.

 

There are two squares side by side. The side of the yellow square is 2. Find the area of the red triangle.

 

 

 

 

Traditional solution

 

Join CE.

 

BG and CE are diagonals of the adjacent squares.

BG//CE.

Now, consider DBGC and D BGE,

The triangles have the same base BG and between the parallel line BG and CE (They have the same height CH)

area of DBGC = area of DBGE

 

\ Area of triangle in red

= area of D BGE

= area of DBGC

= (1/2) BC2 = (1/2)(2)2

= 2

 

 

 

 

 

 

 

“Investigate the extreme” method

 

             Since there is nothing given to the side of the blue square. The question is the same by making the blue square smaller. Consider the extreme case by reducing the blue square to a point, so that point E becomes point C. The area of the triangle in red, that is, DBGE, becomes DBGC at once. Its area is thus 2.

 

 

Problem

 

Imagine the earth is a sphere. A rope A ties around the equator of the earth. Another rope B circulates the equator but is 1 m above the ground.

 

Guess the difference in lengths of the two ropes A and B. Do you think that the difference is longer then the height of the highest building in Hong Kong?  

(radius of Earth = 6 378.1 kilometers)

 

Now do the calculations. Can we use the “investigate the extreme” method?