The question is: **Two 20° involute gears are in mesh. The maximum length of path of contact for two gears to avoid interference is 100 mm. The centre distance between the gears to the closest integer is:** Options: 1. 126 mm 2. 314 mm 3. 354 mm 4. 292 mm ### **Solution Outline** For gears in mesh, the **path of contact** (maximum) is given as 100 mm. For 20° involute gears: \[ \text{Path of contact} = \text{length of line of action} = \sqrt{r_{a1}^2 - r_b^2} + \sqrt{r_{a2}^2 - r_b^2} - (r_1 + r_2)\sin\phi \] But for standard questions: \[ \text{Path of contact} = \text{Centre Distance} \times \sin \phi \] Given the maximum length of path of contact to avoid interference is **100 mm**, and pressure angle \(\phi = 20^\circ\): \[ \text{Centre Distance} = \frac{\text{Path of Contact}}{\sin 20^\circ} \] \[ \sin 20^\circ \approx 0.3420 \] \[ \text{Centre Distance} = \frac{100}{0.3420} \approx 292.4\, \text{mm} \] **Closest integer is: 292 mm** ### **Final Answer:** **Option 4: 292 mm**