Let the length of the first nichrome wire=l then the length of the second nichrome wire =4l,...................................let the cross section area of the first wire =A then,the cross section area of the second wire =3A.....................let resistance of the first wire =R1 and resistance of second wire =R2....then,R1/R2=(p×l/A)/(p×4l/3A)=3/4............now,it is given that resistance of first wire is 5 ohm so,R1/R2=3/4=>5/R2=3/4...=>R2=20/3=6.67 ohm(ans)

**Given information:**
- Resistance of the first nichrome wire: 5 Ω

**To find:**
- Resistance of the second nichrome wire with four times the length and three times the area of cross-section of the first wire.

**Solution:**

**Step 1: Understanding the relationship between resistance, length, and cross-sectional area**

The resistance of a wire depends on its length and cross-sectional area. The formula for resistance is:

Resistance (R) = Resistivity (ρ) * Length (L) / Cross-sectional Area (A)

where ρ is the resistivity of the material, L is the length of the wire, and A is the cross-sectional area of the wire.

**Step 2: Calculating the resistivity of the nichrome wire**

We are given the resistance of the first wire (5 Ω), but we need to find the resistivity (ρ) in order to calculate the resistance of the second wire. To do this, we can rearrange the formula:

Resistivity (ρ) = Resistance (R) * Cross-sectional Area (A) / Length (L)

**Step 3: Calculating the resistance of the second wire**

The second wire has four times the length and three times the cross-sectional area of the first wire. Let's denote the length of the second wire as 4L and the cross-sectional area as 3A.

Using the formula for resistivity, we can calculate the resistivity of the second wire:

Resistivity (ρ2) = 5 Ω * (3A) / (4L)

Now, we can calculate the resistance of the second wire using the formula for resistance:

Resistance (R2) = Resistivity (ρ2) * Length (4L) / Cross-sectional Area (3A)

**Step 4: Simplifying the expression**

Let's simplify the expression for the resistance of the second wire:

Resistance (R2) = (5 Ω * 3A * 4L) / (4L * 3A)

Canceling out common factors, we get:

Resistance (R2) = 5 Ω

**Step 5: Conclusion**

The resistance of the second nichrome wire is also 5 Ω, which is the same as the resistance of the first wire. This means that the resistance of a wire is independent of its length and cross-sectional area as long as the material remains the same.