Masses of 3 wires of same metal are in the ratio 1 : 2 : 3 and their lengths are in the ratio 3 : 2 : 1. The electrical resistances are in ratio

Options

A. `1:1:1`
B. `1:2:3`
C. `9:4:1`
D. `27: 6:1`

Explanation

To find the electrical resistances of the three wires given their mass and length ratios, we can follow these steps:

### Step 1: Understand the relationship between resistance, mass, and length
The resistance \( R \) of a wire is given by the formula:
\[ R = \frac{\rho l}{A} \]
where:
- \( R \) is the resistance,
- \( \rho \) is the resistivity of the material,
- \( l \) is the length of the wire,
- \( A \) is the cross-sectional area.

### Step 2: Express the cross-sectional area in terms of mass and length
The mass \( m \) of the wire can be expressed as:
\[ m = \rho \cdot V = \rho \cdot A \cdot l \]
From this, we can express the area \( A \) as:
\[ A = \frac{m}{\rho l} \]

### Step 3: Substitute the expression for area into the resistance formula
Substituting \( A \) into the resistance formula gives:
\[ R = \frac{\rho l}{\frac{m}{\rho l}} = \frac{\rho^2 l^2}{m} \]
This shows that resistance \( R \) is proportional to the square of the length \( l^2 \) and inversely proportional to the mass \( m \):
\[ R \propto \frac{l^2}{m} \]

### Step 4: Set up the ratios based on given data
Given the ratios:
- Masses of the wires: \( m_1 : m_2 : m_3 = 1 : 2 : 3 \)
- Lengths of the wires: \( l_1 : l_2 : l_3 = 3 : 2 : 1 \)

We can express the lengths and masses as:
- \( m_1 = 1k, m_2 = 2k, m_3 = 3k \)
- \( l_1 = 3m, l_2 = 2m, l_3 = 1m \)

### Step 5: Calculate the resistances in terms of these ratios
Using the relationship \( R \propto \frac{l^2}{m} \):
- For wire 1:
\[ R_1 \propto \frac{(3m)^2}{1k} = \frac{9m^2}{k} \]
- For wire 2:
\[ R_2 \propto \frac{(2m)^2}{2k} = \frac{4m^2}{2k} = \frac{2m^2}{k} \]
- For wire 3:
\[ R_3 \propto \frac{(1m)^2}{3k} = \frac{1m^2}{3k} \]

### Step 6: Write the ratios of the resistances
Now we can write the ratios of the resistances:
\[ R_1 : R_2 : R_3 = \frac{9m^2}{k} : \frac{2m^2}{k} : \frac{1m^2}{3k} \]
To simplify, we can eliminate \( m^2/k \):
\[ R_1 : R_2 : R_3 = 9 : 2 : \frac{1}{3} \]
To make the third term comparable, we can multiply all terms by 3:
\[ R_1 : R_2 : R_3 = 27 : 6 : 1 \]

### Final Answer
Thus, the electrical resistances are in the ratio:
\[ \boxed{27 : 6 : 1} \]