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

Let the radii of the sphere be `R,R+a, R+2a` and `R +3a` where `a` is a constant and the specific heat copacites be `s,sr,sr^(2)` and `sr^(3)` where `r` is another constant.
given
`(("heat capacity of" D)/("heat capacity of" B)) : (("heat capacity of" C)/("heat capacity of" A)) = 8:27`
or `[((R+3a)^(3)sr^(3))/((R+a)^(3)sr)] : [((R+2a)^(3)sr^(2))/(Rs)] = 8.27`
or , `(1+(2a)/(R+a)) : (1+(2a)/(R)) = 2:3`
or `(2a)/(R+a) : (2a)/(R) = 1:2`
or `R=a`
`(m_2)/(m_1) = ((4//3)pi(R+R)^(3)rho)/((4//3)pi(R)^(3)rho) = 8/1`.