9. Figure 4.83 shows a tapered gauge (plug) that is checked/tested for accuracy by means of rwo precion balls. 9.1 Explain the steps you would take to use the precision balls to determine the included angle of the ring gauge. 9.2 Make a large and clear copy of the diagram. Use the dimensions given in the diagram to calculate the included angle of the ring gauge.

To solve the problem, we will break it down into two parts as requested: ### 9.1 Steps to Determine the Included Angle of the Ring Gauge 1. **Setup the Precision Balls**: Place the two precision balls in the tapered gauge. Ensure that they are positioned at the widest part of the gauge where the taper begins. 2. **Measure the Distance Between the Balls**: Use a caliper or a similar measuring tool to accurately measure the distance between the centers of the two precision balls. Let's denote this distance as \( d \). 3. **Measure the Diameter of the Balls**: Measure the diameter of one of the precision balls. Denote this diameter as \( D \). 4. **Calculate the Radius of the Balls**: The radius \( r \) of the balls can be calculated as: \[ r = \frac{D}{2} \] 5. **Determine the Geometry**: The included angle \( \theta \) of the ring gauge can be determined using the geometry of the situation. The distance between the centers of the balls forms a triangle with the radius of the balls. 6. **Use Trigonometry**: The relationship between the distance \( d \), the radius \( r \), and the included angle \( \theta \) can be expressed using the tangent function: \[ \tan\left(\frac{\theta}{2}\right) = \frac{r}{\frac{d}{2}} \] Rearranging gives: \[ \theta = 2 \cdot \tan^{-1}\left(\frac{r}{\frac{d}{2}}\right) \] 7. **Calculate the Included Angle**: Substitute the values of \( r \) and \( d \) into the equation to find \( \theta \). ### 9.2 Diagram and Calculation Since I cannot create visual diagrams, I will guide you on how to create a clear copy of the diagram based on the dimensions provided in the original figure. 1. **Draw the Tapered Gauge**: Sketch the tapered gauge shape, ensuring to represent the taper accurately. 2. **Indicate the Precision Balls**: Draw two circles representing the precision balls at the widest part of the gauge. 3. **Label Dimensions**: Clearly label the distance \( d \) between the centers of the balls and the diameter \( D \) of the balls. 4. **Calculate the Included Angle**: Now, let's proceed to calculate the included angle using the dimensions provided in the diagram. Please provide the specific dimensions (values for \( d \) and \( D \)) so that I can perform the calculation.