The height (in mm) for a 125 mm sine bar to measure a taper of 27°32’ on a flat work piece is __________ (correct to three decimal places).

The correct answer is 57.782.

To find the required height of the slip gauges to set a sine bar for a given angle, we can use the fundamental trigonometric relationship of a sine bar setup:
$\sin \left(\theta \right)=\frac{H}{L}$
where:
- $\theta$ is the angle of the taper to be measured,
- $H$ is the height of the slip gauges (in mm),
- $L$ is the length of the sine bar (in mm).

First, we convert the angle $\theta ={27}^{\circ }32\prime$ into decimal degrees:
$\theta =27+\frac{32}{60}\approx 27.5333°$

Next, we substitute the given length of the sine bar ($L=125\text{mm}$) and the decimal angle into the formula to solve for the height $H$:
$H=L\times \sin \left(\theta \right)$
$H=125\times \sin (27.5333°)$
$H\approx 125\times 0.462259$
$H\approx 57.7824\text{mm}$

Rounding to three decimal places as specified by the question, we get:
$H\approx 57.782\text{mm}$