Question

The resistance of a wire is proportional to its length and inversely proportional to the square of its radius. Two wires of the same material have the same resistance and their radii are in the ratio \(9:8\). If the length of the first wire is 162 cm, find the length of the other.

• A. 64 cm
• B. 120 cm
• C. 128 cm
• D. 132 cm

Hint:

For constant resistance, \(L \propto r^2\). Use given radius ratio to find the missing length.

Answer 1

The Correct Option is C

Given \( R \propto \frac{L}{r^2} \). For two wires with the same \(R\): \[ \frac{L_1}{r_1^2} = \frac{L_2}{r_2^2} \] Given \(r_1 : r_2 = 9 : 8\): \[ \frac{L_1}{(9)^2} = \frac{L_2}{(8)^2} \quad \Rightarrow \quad \frac{L_1}{81} = \frac{L_2}{64} \] Cross-multiplying: \[ 64L_1 = 81L_2 \] Substitute \(L_1 = 162\): \[ 64 \times 162 = 81 L_2 \] \[ L_2 = \frac{10368}{81} = 128 \ \text{cm} \] Thus \({128 \ \text{cm}}\) is correct.