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# geometry

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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?

Mar 29, 2024

#1
+129803
+1

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

Mar 29, 2024