Grounded Sphere
Sphere with a diameter is resting on a horizontal plane. A and B are points on the surface of the sphere. Point A is 9 cm off the ground, the straight line distance is , and the tangent of the angle is 7. Find the difference between the smallest and the largest possible values of .
Image of sphere from kingofwallpapers.com.
The answer is 1680.
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1 solution
This is a 2D problem in a 3D disguise. The center of the smallest possible sphere is directly below the line connecting A and B, while the center of the largest possible sphere is directly above that line. We can picture the plane in which all of this is happening and set up a coordinate system in it.
The coordinates of point A are (0, 9). The ratio of DB to DA, which is the tangent of angle BAC, is 7. If we set DB = 7 and DA = 1 that would be the easiest way to satisfy this requirement. This will give us the A B = 5 0 as required. Coordinates of point B are therefore (7, 8).
The center of the sphere needs to be a distance R from: (1) point A, (2) point B, and (3) the ground. This gives us three equations:
x 2 + ( y − 9 ) 2 = R 2
( x − 7 ) 2 + ( y − 8 ) 2 = R 2
y = R
Solving for R we get two answers: R = 5 and R = 8 4 5 . The difference of the diameters is therefore quite a large figure, namely 1680.