Figure 4.81 shows two angled faces that have to be tested for correctness with the aid of rollers, 15 mm in diameter. Calculate the distance M between the rollers for the angles to be correctly machined.

The diameter of the rollers is $d = 15 \text{ mm}$.
The radius of each roller is $r = \frac{d}{2} = \frac{15}{2} = 7.5 \text{ mm}$.
The bottom width of the groove is $80 \text{ mm}$.

The diagram shows an internal angle of $130^\circ$ between the top horizontal line and the left inclined face.
The angle of the left inclined face with the horizontal, $\theta_L$, is $180^\circ - 130^\circ = 50^\circ$.

For a roller tangent to a horizontal surface and an inclined surface, the horizontal distance from the corner (where the two surfaces meet) to the center of the roller is given by $x = r \cot(\theta/2)$, where $\theta$ is the angle of the inclined surface with the horizontal.
For the left roller:

The diagram shows an internal angle of $112^\circ$ between the top horizontal line and the right inclined face.
The angle of the right inclined face with the horizontal, $\theta_R$, is $180^\circ - 112^\circ = 68^\circ$.

For the right roller:

The distance $M$ is the horizontal distance between the centers of the rollers. This can be found by subtracting the horizontal distances $x_1$ and $x_2$ from the total bottom width of $80 \text{ mm}$.

Rounding to two decimal places:

The final answer is $52.80 \text{ mm}$.