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| The area not reachable by the goat is the difference in area between the segments | |||||||||||
| subtended by the chord XY in the circles radius R and r. | |||||||||||
| Area in segment subtended in circle radius R is | |||||||||||
| (2A/2PI)*PI*R*R - R*R*Cos(A)*Sin(A) | |||||||||||
| Similarly area in segment subtended in circle radius r is | |||||||||||
| (4A/2PI)*PI*r*r - r*r*Cos(2A)*Sin(2A) | |||||||||||
| So we want therefore that (the area that the goat can't reach): | |||||||||||
| (4A/2PI)*PI*r*r - r*r*Cos(2A)*Sin(2A) - (2A/2PI)*PI*R*R + R*R*Cos(A)*Sin(A) =PI*r*r/2 | |||||||||||
| Now notice that | |||||||||||
| R=2*r*Cos(A) | |||||||||||
| Substituting for R in (1) we get | |||||||||||
| (4A/2PI)*PI*r*r - r*r*Cos(2A)*Sin(2A) - (2A/2PI)*PI*4*r*r*Cos(A)*Cos(A) + 4*r*r*Cos(A)*Cos(A)*Cos(A)*Sin(A) | |||||||||||
| = PI*r*r/2 | |||||||||||
| Dividing by r*r sounds like a good idea | |||||||||||
| 2A - Cos(2A)*Sin(2A) - 4*A*Cos(A)*Cos(A) + 4*Cos(A)*Cos(A)*Cos(A)*Sin(A) - PI/2 = 0 | |||||||||||
| Using the goal seeker in Excel gives A as | 0.95282 | radians | |||||||||
| and the ratio of the radiii (R/r) = 2 Cos (A)= | 1.158774 | ||||||||||
| Goal Seeker...... | |||||||||||
| A | Function | R/r | |||||||||
| (Radians) | Evaluation | ||||||||||
| 0.95282 | -0.000100728765
|
1.158774 | |||||||||