Calculate Staff Intercept ($S$)

The staff intercept $S$ is the difference between the top hair reading ($S_{top}$) and the bottom hair reading ($S_{bottom}$).

$S = S_{top} - S_{bottom} = 3.0\text{ m} - 1.0\text{ m} = 2.0\text{ m}$

Apply Vertical Distance Formula

For an inclined line of sight, the formula for vertical distance $V$ is $V = k \cdot S \cdot \sin(\theta) \cdot \cos(\theta) + c \cdot \sin(\theta)$, which simplifies using trigonometric identity $\sin(2\theta) = 2 \sin(\theta) \cos(\theta)$.

$V = k \cdot S \cdot \frac{\sin(2\theta)}{2} + c \cdot \sin(\theta)$

Substitute Values and Solve

Substitute $k = 100$, $S = 2.0\text{ m}$, $\theta = 10°$, and $c = 0$ into the formula.

$V = 100 \times 2.0 \times \frac{\sin(20°)}{2} = 100 \times \sin(20°)$

Final Numerical Calculation

Since $\sin(20°) \approx 0.3420201$, we multiply by $100$ to get the vertical distance.

$V = 100 \times 0.3420201 = 34.202\text{ m}$