A spherical shell of outer radius R and inner radius R / 2 contains a solid sphere of radius R / 2 (see figure). The density of the material of the solid sphere is and that of the shell is / 2. What is the average mass density of the larger sphere thus formed?
B. 9 / 16
aligned & Average density = Total mass Total volume \\ & = _1 V_1+ _2 V_2 V \\ & = 4 3 ( R 2 )^3+ 2 [ 4 3 R ^3- 4 3 ( R 2 )^3 ] 4 3 R ^3 \\ & = 4 3 R ^3 8 + 2 [ 4 3 R ^3- 4 3 R ^3 8 ] 4 3 R ^3 \\ & = 8 + 2 [1- 1 8 ] \\ & = 8 + 2 [ 7 8 ] \\ & = 8 + 7 16 \\ & = 2 +7 16 = 9 16 . aligned
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