+1234 Rose has a spherical plum of radius $2$ and a spherical watermelon of radius $8$. She builds a glass sphere around the two fruits to contain them, making the sphere as small as possible. When she has done this, she places a golf ball inside the glass sphere that is tangent to the plum and watermelon. What is the largest possible radius of the golf ball?
+129845 Something like this
Law of Cosines { cos (angle BDA) = -cos (angle CDA) }
BA^2 = BD^2 + DA^2 -2 (BD * DA) cos (BDA)
AC^2 = DC^2 + DA^2 - 2 (BD * DA) (CDA)
(8 + r)^2 = 2^2 + (10 -r)^2 -2 [ 2 (10-r) ] (-cos (CDA))
(2 + r^2) = 8^2 + (10 - r)^2 - 2 [ 8 (10-r)] (cos (CDA))
(8 + r)^2 = 2^2 + (10 -r)^2 +2 [ 2 (10-r) ] (cos (CDA))
(2 + r^2) = 8^2 + (10 - r)^2 - 2 [ 8 (10-r)] (cos (CDA))
Equating Cosines
[(8 + r)^2 - 2^2 -(10 - r)^2] / [ 4 (10 -r))] = [ (2 + r)^2 - 8^2 - (10-r)^2] / [ -16 (10 -r) ]
[ r^2 + 16r + 64 - 4 - r^2 + 20r -100 ] / 4 = [ r^2 + 4r + 4 - 64 - r^2 + 20r - 100] / -16
[ 36r -40 ] / 4 = [ 24r -160 ] / -16
- 4 [ 36r -40 ] = 24r - 160
-144r + 160 = 24r - 160
24r + 144r = 320
r [ 168 ] = 320
r = 320 / 168 = 40/21 ≈ 1.9
