61199829-131c-4509-bf1d-1f0540966fd8 50 3 5 decimal

&[DATE] &[TIME] - &[FILENAME]

I hate FindRoot ! f init GivenFind f init FindRoot :

http://mathworld.wolfram.com/GoatProblem.html

Tieing a goat to a point on the interior of the fence with radius a=1 using a chain of length L=r, consider the length of chain that must be used in order to allow the goat to graze exactly one half the area of the field. The answer is obtained by using the equation for a circle-circle intersection ... which cannot be solved exactly, but which has approximate solution: L= 1.15872847

R 1 :

The solve bloc fails miserably ...  use "solve" instead. f 212 / R 2^* 2 L 2 R * / asin * 2 L 2 R * / asin * sin - (* (* L 2^L 2 R * / acos * + π R 2^* 2 / ≡ :

init L 1.1401 ≡ :

f init GivenFind 1.15868309

R L Goat 212 / R 2^* 2 L 2 R * / asin * 2 L 2 R * / asin * sin - (* (* L 2^L 2 R * / acos * + :

x c 10 x ≤ (x 2 R * ≤ (& cases :

R x Goat x c * 1.1587 π R 2 ^ * 2 / + 15 red 1 5 mat 2 1 sys

... a more mathematical competitive solution R L Goat π R 2^* 2 / - 0 ≡ L 1 1.2 solve 1.15872845

Solution: Valery Okhov ... Mathsoft Collaboratory

21 x 2^2 / - asin * 2 x 2^* x 2 / asin * + 4 x 2^* x 4^- sqrt + π x 2^* - 0 ≡

x D R / ≡

Carlos:Smath Forum

21 x 2^2 / - asin * 2 x 2^* x 2 / asin * + 4 x 2^* x 4^- sqrt + π x 2^* - 0 ≡ x 02 solve 1.158728473

==================================================================================

xo 1.15872847 :

shift 1 1.15872847 - : 0.159 -

x C1 xo 2^x shift - xo - (2^- sqrt ± :

x C2 1 x 2^- sqrt ± :

X x C1 1 el x C2 1 el - 0 ≡ x 0 0.5 solve : Y 0 C1 : 21 mat 0.3287 0.5854 0.5854 - 21 sys 21 mat

data spec pts i 1 data rows range p i el a b c spec : for Unest 'p' v p 1 el : j 2 p rows range v v p j el stack : for data v augment 51 line :

x C1 x C2 2 1 sys

#A0A0A0

16016016013 mat

x y1 x C1 1 el : x y2 x C2 1 el : 21 sys

X 0.3287 :

Y 0.5854 :

Density of random N 500 :

A function that tests an arbitrary point 
for its membership in the critical region. x y Test y x y1 + 0 ≥ (y x y1 - 0 ≤ (* :

xmax 0 : xmin shift : 21 line

ymax 1 : ymin 1 - : 21 line

space_x xmax xmin - (: space_y ymax ymin - (: 21 line

A routine that finds an array of N random 
pairs in the critical region. Pts i 1 N range x 1 Random space_x * xmin + (1 el : y 1 Random space_y * ymin + (1 el : 1 x y Test - x 1 Random space_x * xmin + (1 el : y 1 Random space_y * ymin + (1 el : 21 line while P i 1 el x : P i 2 el y : 51 line for P 21 line :

a b c spec . 5 blue 13 mat : pts Pts a b c spec pts : λ1 pts a b c spec pts : 31 sys

The routine 'Pts' is a compressor. It maintians density 'N' within the critical region of variable area.

OBSERVE: 'Pts' may fail on too many decimals [X, Y]

Testing parenthood with y2 y x y2 + 0 ≥ (y x y2 - 0 ≤ (*

xmax X : xmin 0 : 21 line

ymax 1 : ymin Y : 21 line

λ3 is a combination of y1 & y2 ... λ4 follows from λ3 x y Test y x y1 + 0 ≥ (y x y1 - 0 ≤ (* y x y2 + 0 ≥ (y x y2 - 0 ≤ (* - :

space_x xmax xmin - (: space_y ymax ymin - (: 21 line

A routine that finds an array of N random 
pairs in the critical region. Pts i 1 N range x 1 Random space_x * xmin + (1 el : y 1 Random space_y * ymin + (1 el : 1 x y Test - x 1 Random space_x * xmin + (1 el : y 1 Random space_y * ymin + (1 el : 21 line while P i 1 el x : P i 2 el y : 51 line for P 21 line :

Isolate Pts to avoid random interference

CWD C:\SmathFile CurrentDirectory :

Pts CWD Goat wfile 1

Pts CWD Goat rfile :

Plot x C1 x C2 λ1 λ3 λ4 51 sys

a b c spec . 5 #A0A0A0 13 mat : pts3 Pts a b c spec pts : λ3 pts3 a b c spec pts : pts4 Pts 1 col Pts 2 col - augment : λ4 pts4 a b c spec pts : 51 sys

x C1 x C2 λ1 λ3 λ4 5 1 sys

xC yC R circle Δθ π 192 / : k 12 π * Δθ / 1 + (range x k el R Δθ k * cos * eval : y k el R Δθ k * sin * eval : 21 line for xC x + eval yC y + eval augment 31 line :

Diameter C1
on plot @ x=0 x f1 x 1 x 1 - 22 mat :

length arrow
angle sharpness H.L 0.25 : H.W 0.0625 : 21 sys

base length unary circle L.v 1 :

SP.x SP.y L.v α Arrow 00 L.v 0 L.v H.L - H.W 2 / L.v H.L - H.W 2 / - L.v 052 mat α cos α sin α sin - α cos 22 mat * (SP.x SP.y SP.x SP.y SP.x SP.y SP.x SP.y SP.x SP.y 52 mat + :

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 31 line for data r3 r4 r5 augment 21 line :

SP.x 0 :

SP.y 0 :

L.v 1 :

C1 001 circle : C2 10 1.15872847 circle : C1 C1 . 5 blue plot : C2 C2 . 9 red plot : 41 sys

Text 0.5 0.5 D 15 red 0.35 0.25 - R 15 blue 25 mat :

Project SP.x SP.y L.v 310 ° * Arrow 1 SP.y 1.15872847 125 ° * Arrow C1 C2 Text 0 f1 61 sys :

Project λ1 λ3 λ4 4 1 sys