If the North American Plate moves relative to the Pacific Plate along a transform fault at a rate of $50 \text{ mm/year}$, how far will it move over a period of $1 \text{ million years}$?

Two tectonic plates are converging at a rate of $3.5 \text{ cm/year}$. Geologists have determined through paleomagnetic dating that the two plates were originally separated by a distance of $1,400 \text{ km}$. Assuming a constant convergence rate, how long ago did these two points begin to converge?

A new volcanic island is forming over a mantle hotspot. The tectonic plate above the hotspot is moving northwest at a velocity of $7 \text{ cm/year}$. Over a period of $2.5 \text{ million years}$, a chain of volcanic islands is formed. Given that the relative orientation of the plate motion with respect to a north-south grid is exactly $45^{\circ}$ West of North, calculate the total displacement in both the North and West directions.

During the preliminary site investigation for a proposed concrete gravity dam, boreholes revealed a complex stratigraphic sequence. The top $15 \text{ m}$ consist of horizontally bedded, highly permeable sandstone. Below this lies an angular unconformity, where the underlying rock consists of steeply dipping, highly foliated schist. The project requires a robust, impermeable foundation to prevent massive seepage and catastrophic failure.

A high-rise commercial building is planned in a region with a moderate geothermal gradient ($25^{\circ}\text{C/km}$). The building requires extensive deep foundations (piles) extending $40 \text{ m}$ into the bedrock to support structural loads. The engineering team proposes integrating closed-loop geothermal heat exchangers directly into the concrete foundation piles (energy piles) to provide sustainable heating and cooling for the building.

A major transportation corridor is being planned across a region known to be located near an active convergent plate boundary, where an oceanic plate is subducting beneath a continental plate. The route must cross several major river valleys and mountain passes.