The masses of the three wires of copper are in the ratio `5:3:1` and their lengths are in the ratio `1:3:5`. The ratio of their electrical resistances is

Options

A. `5:3:1`
B. `sqrt(125):15:1`
C. `1:15:125`
D. `1:3:5`

Answer

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

### Step 1: Understand the relationship between resistance, length, and volume
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 (constant for copper),
- \( L \) is the length of the wire,
- \( A \) is the cross-sectional area.

### Step 2: Express the area in terms of mass and density
The volume \( V \) of the wire can be expressed as:
\[ V = \frac{m}{\rho} \]
where \( m \) is the mass of the wire. The cross-sectional area \( A \) can be expressed as:
\[ A = \frac{V}{L} = \frac{m}{\rho L} \]
Substituting this into the resistance formula gives:
\[ R = \frac{\rho L}{\frac{m}{\rho L}} = \frac{\rho^2 L^2}{m} \]
This shows that resistance is directly proportional to the square of the length and inversely proportional to the mass.

### Step 3: Set up the ratios
Given:
- Mass ratio of the wires: \( 5:3:1 \)
- Length ratio of the wires: \( 1:3:5 \)

Let’s denote the lengths as \( L_1 = 1 \), \( L_2 = 3 \), and \( L_3 = 5 \). The corresponding masses are \( m_1 = 5k \), \( m_2 = 3k \), and \( m_3 = 1k \) for some constant \( k \).

### Step 4: Calculate the resistance ratios
Using the derived relationship for resistance:
\[ R \propto \frac{L^2}{m} \]
We can write the resistance for each wire:
- For wire 1:
\[ R_1 \propto \frac{(1)^2}{5} = \frac{1}{5} \]
- For wire 2:
\[ R_2 \propto \frac{(3)^2}{3} = \frac{9}{3} = 3 \]
- For wire 3:
\[ R_3 \propto \frac{(5)^2}{1} = \frac{25}{1} = 25 \]

### Step 5: Write the resistance ratio
Thus, the ratio of the resistances \( R_1 : R_2 : R_3 \) is:
\[ R_1 : R_2 : R_3 = \frac{1}{5} : 3 : 25 \]

### Step 6: Normalize the ratio
To express this in a simpler form, we can multiply each term by 5 (the common denominator):
\[ R_1 : R_2 : R_3 = 1 : 15 : 125 \]

### Final Answer
The ratio of the electrical resistances of the three wires is:
\[ \boxed{1 : 15 : 125} \]