PUZZLET 97 - THE SOLUTION
| The
heart of this problem is to be able to express the ratio of the area of
the triangle to that of the enclosed circle in terms of the triangle's
base angle. Here's one way to develop a suitable formula: |
 |
| With
the formula in our metaphorical toolkit, we can write some code. |
| ' Puzzlet #097 ' A circle is enclosed by an isosceles ' triangle so that the circumference is ' just touching the two sides and base. ' What base angle gives a ratio of the ' the triangle's area to the area of ' the circle equal to 2? 3? 4? 5? Work ' to an accuracy of four decimal places. ' By Dave Ellis. declare Func (angle: float) declare RadToDeg (rads: float) declare DegToRad (degs: float) declare PrintResult() def ave, fAve, max, min, pi: float def ratio: int pi = 3.141593 color 14, 0 openconsole print "Calculating ...": print print "Ratio Angle (degs)": print for ratio = 2 to 5 max = DegToRad(89) min = DegToRad(1) do ave = (max + min)/2 fAve = Func(ave) if fAve > ratio max = ave else min = ave endif until abs(fAve - ratio) < .00001 PrintResult() next ratio print: print "Press a key ... ", do: until inkey$ <> "" closeconsole end sub Func(angle) ' returns the ratio of the triangle's area ' to that of the circle as a function of the ' triangle's base angle def numerator, denominator: float numerator = tan(angle) denominator = pi*(tan(angle/2))^2 return numerator/denominator sub DegToRad (degs) ' converts degrees to radians def rads: float rads = degs * pi/180 return rads sub RadToDeg (rads) ' converts radians to degrees def degs: float degs = rads * 180/pi return degs sub PrintResult ' pretty printer color 11,0 print " ", ratio, print " ", print using ("##.####", RadToDeg(ave)) color 14,0 return |
| PROGRAM NOTES The program is very simple once you have the formula. An outer loop iterates variable ratio over the target values of 2 to 5 to provide the solutions required for those values. Within that an inner DO loop applies the bisection method. This has been used in puzzlets many times before. If you would like more information, please refer back using the archives or e-mail me directly. At each iteration of the bisection method, the formula is applied when the function Func() is called, since that's where the formula is located. The result is returned as an angle. If the resolution is insufficient, the DO loop is executed again with the latest values, otherwise the program prints the result and quits. |
| When
you run the program, you'll see the inset results printed out onto your
computer's screen. Larger ratios are successively closer to 90o, and the angular differences get smaller. |
|
Calculating
... Ratio Angle (degs) 2 74.7350 3 82.0100 4 84.4888 5 85.7807 Press a key ... |