A sedimentary rock bed has a true dip ($\alpha$) of $30^{\circ}$ and a strike of $N45^{\circ}E$. A road cut is planned to intersect this bed along a trend of $N90^{\circ}E$ (due East). What is the apparent dip ($\alpha'$) of the bed as it will appear on the face of the road cut?

An engineer observes a fault plane exposed on a vertical cliff face that trends $N15^{\circ}W$. On this cliff face, the fault appears to dip at an angle of $40^{\circ}$ (this is the apparent dip, $\alpha'$). Geological mapping indicates the true strike of the fault is $N45^{\circ}W$. What is the true dip ($\alpha$) of the fault plane?

A coal seam has a true dip of $25^{\circ}$ and strikes due North ($N0^{\circ}E$). A mining tunnel needs to be excavated through the seam. The mining equipment can only operate on an incline of $10^{\circ}$ or less. What is the maximum angle ($\beta$) the tunnel can deviate from the strike direction to maintain an apparent dip ($\alpha'$) of exactly $10^{\circ}$?

A $5 \text{ km}$ railway tunnel is proposed to cut straight through a mountain range. Detailed geological mapping reveals the mountain is primarily a large, asymmetrical anticline fold composed of alternating layers of hard, brittle limestone and soft, impermeable shale.

A massive concrete gravity dam is planned across a deep river gorge. Geological surveys indicate a major thrust fault (a low-angle reverse fault) intersects the gorge directly at the proposed dam foundation site.