Question

An underground vertical dyke is intercepted by an inclined borehole as shown in the figure below. The length of the dyke core intercepted by the borehole is 4m. If the true thickness of the dyke is 2m, the inclination of the borehole from the vertical is __________ degrees. (In integer)

Answer 1

Correct Answer: 30

Given:
True thickness of vertical dyke = \(t=2\ \text{m}\). 
Intercepted core length along the borehole = \(L=4\ \text{m}\).
Borehole is inclined from the vertical by angle \(\theta\). We assume the borehole intersects the dyke so that its horizontal projection is normal to the dyke (i.e. it cuts across the thickness directly). 
Geometry & formula:
Let \(\mathbf{n}\) be the unit normal to the (vertical) dyke (horizontal direction). If the borehole direction makes angle \(\theta\) from the vertical, its horizontal component (towards the dyke normal) has magnitude \(\sin\theta\). The length of core inside the dyke equals the true thickness divided by the component of the borehole direction along the normal: \[ L=\frac{t}{\sin\theta}. \] Compute: \[ 4=\frac{2}{\sin\theta}\quad\Rightarrow\quad \sin\theta=\frac{2}{4}=0.5. \] Thus \[ \theta=\arcsin(0.5)=30^\circ. \] Answer (integer): \(\boxed{30^\circ}\)