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Inconvenient points (Posted on 2021-08-23) Difficulty: 3 of 5
Two vertical poles stand 8.4m apart. AA' is 4.4m high, BB' is 3.1m high (A' and B' lying on the ground). A point P on the ground is defined to be a “convenient point”, if the viewing angle of points A and B from P is an acute one. If you move away from the poles, you can certainly find convenient points. There is a region of points on the ground, however, where all points are inconvenient.

Find the area of this region of inconvenient points.

No Solution Yet Submitted by Danish Ahmed Khan    
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soln Comment 2 of 2 |
We put pole base A' at the origin and pole base B' at b=8.4m on the x-axis.
The border of the inconvenient region is the locus of points P where the opening angle theta of vectors V1=(P-A) and V2=(P-B) is 90 degrees. Within this border the angle becomes obtuse.

V1 = (x, y, 0) - (0, 0, z1) = (x  , y, -z1)         z1 = 4.4m
V2 = (x, y, 0) - (b, 0, z2) = (x-b, y, -z2)        z2 = 3.1m

cos(theta) = V1 dot V2 / (mag V1) (mag V2)
cos(90) = 0 = V1 dot V2 

x^2 - bx + y^2 + z1 z2 = 0

(x-b/2)^2 - (b/2)^2 + y^2 + z1 z2 = 0

(x-b/2)^2 + y^2 = (b/2)^2 -z1 z2  
which is a circle of radius sqrt[ (b/2)^2- z1 z2] = 2m
centered at b/2 = 4.2m
(x-4.2)^2 - y^2 = 4
Area = 4 pi m^2

Edited on August 23, 2021, 11:21 pm
  Posted by Steven Lord on 2021-08-23 12:18:20

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