Four spheres `A`, `B`, `C` and `D` have their radii in arithmetic progression and the specific heat capacities of their substances are geometric progression. If the ratios of heat capacities of `D` and `B` to that of `C` and `A` are as `8:27`. The ratio of masses of `B` and `A` is: (assume same density for all spheres)

Options

A. ` 8 : 1`
B. `4 : 1`
C. `1 : 8`
D. `1 : 4`

Answer

A

Explanation

`(C_(D))/(C_(B))//(C_(C))/(C_(A))=(8)/(27)or,(C_(D)C_(A))/(C_(B)C_(C))=(8)/(27)rArr(m_(D)S_(D).m_(A)S_(A))/(m_(B)S_(B).m_(C)S_(C))=(8)/(27)or,(m_(A)m_(D))/(m_(B)m_(C))=(8)/(27)`
or `((r_(A))^(3)(r_(A)+3(r_(B)-r_(A)))^(3))/((r_(B))^(3)(r_(B)+(r_(B)-r_(A)))^(3))=(8)/(27)or,(r_(A)(3r_(B)-2r_(A)))/(r_(B)(2r_(B)-r_(A)))=(2)/(3)3(3r_(A)r_(B)-2r_(A)^(2))=(2r_(B)^(2)-r_(B)r_(A))^(2)`
`9r_(A)r_(B)-6r_(A)^(2)=4r_(B)^(2)-2r_(B)r_(A) " " 4r_(B)^(2)+6r_(A)^(2)-11r_(A)r_(B)=0`
`4r_(B)(r_(B)-2r_(A))+3r_(A)(2r_(A)-r_(B))=0or,(4r_(B)-3r_(A))(r_(B)-2r_(A))=0`
`or,r_(B)=2r_(A)" as "r_(B)gtr_(A),(m_(B))/(m_(A))=(8)/(1)`