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`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Define the radii and specific heat capacities
Let the radii of the spheres \( A, B, C, D \) be represented as:
- \( r_A = r \)
- \( r_B = r + n \)
- \( r_C = r + 2n \)
- \( r_D = r + 3n \)

The specific heat capacities of the substances are in geometric progression. Let:
- \( c_A = S_0 \)
- \( c_B = m S_0 \)
- \( c_C = m^2 S_0 \)
- \( c_D = m^3 S_0 \)

### Step 2: Calculate the masses of the spheres
Since the density (\( \rho \)) is the same for all spheres, the mass of each sphere can be calculated using the formula for the volume of a sphere \( V = \frac{4}{3} \pi r^3 \):
- \( m_A = \rho \cdot V_A = \rho \cdot \frac{4}{3} \pi r^3 \)
- \( m_B = \rho \cdot V_B = \rho \cdot \frac{4}{3} \pi (r + n)^3 \)
- \( m_C = \rho \cdot V_C = \rho \cdot \frac{4}{3} \pi (r + 2n)^3 \)
- \( m_D = \rho \cdot V_D = \rho \cdot \frac{4}{3} \pi (r + 3n)^3 \)

### Step 3: Write the heat capacities
The heat capacity \( H \) is given by the product of mass and specific heat capacity:
- \( H_A = m_A c_A = \left(\frac{4}{3} \pi r^3 \rho\right) S_0 \)
- \( H_B = m_B c_B = \left(\frac{4}{3} \pi (r + n)^3 \rho\right) m S_0 \)
- \( H_C = m_C c_C = \left(\frac{4}{3} \pi (r + 2n)^3 \rho\right) m^2 S_0 \)
- \( H_D = m_D c_D = \left(\frac{4}{3} \pi (r + 3n)^3 \rho\right) m^3 S_0 \)

### Step 4: Set up the ratio of heat capacities
According to the problem, the ratio of the heat capacities of \( D \) and \( B \) to that of \( A \) and \( C \) is given as:
\[ \frac{H_D + H_B}{H_A + H_C} = \frac{8}{27} \]

### Step 5: Substitute the expressions for heat capacities
Substituting the expressions we derived:
\[ \frac{\left(\frac{4}{3} \pi (r + 3n)^3 \rho m^3 S_0\right) + \left(\frac{4}{3} \pi (r + n)^3 \rho m S_0\right)}{\left(\frac{4}{3} \pi r^3 \rho S_0\right) + \left(\frac{4}{3} \pi (r + 2n)^3 \rho m^2 S_0\right)} = \frac{8}{27} \]

### Step 6: Cancel common terms
Cancel out the common terms \( \frac{4}{3} \pi \rho S_0 \):
\[ \frac{(r + 3n)^3 m^3 + (r + n)^3 m}{r^3 + (r + 2n)^3 m^2} = \frac{8}{27} \]

### Step 7: Solve the equation
Cross-multiplying gives:
\[ 27\left[(r + 3n)^3 m^3 + (r + n)^3 m\right] = 8\left[r^3 + (r + 2n)^3 m^2\right] \]

### Step 8: Find the ratio of masses
To find the ratio of the masses of \( B \) and \( A \):
\[ \frac{m_B}{m_A} = \frac{(r + n)^3}{r^3} \]

### Step 9: Substitute \( n = r \) (from the quadratic equation derived)
If we assume \( n = r \):
\[ \frac{(r + r)^3}{r^3} = \frac{(2r)^3}{r^3} = \frac{8r^3}{r^3} = 8 \]

Thus, the ratio of the masses of \( B \) and \( A \) is:
\[ \frac{m_B}{m_A} = 8:1 \]

### Final Answer
The ratio of the masses of \( B \) and \( A \) is \( 8:1 \).