Optimizing Hiking Routes on Mountain Terrain

You work as a trail guide at a state park, helping hikers plan safe and efficient routes across mountainous terrain. The park uses a mathematical model to represent elevation across the hiking area, and understanding how elevation changes in different directions helps you advise hikers on the steepest climbs, gentlest descents, and paths that maintain constant elevation. Your task is to analyze the terrain using directional derivatives and gradients to answer questions about rate of elevation change, optimal travel directions, and level paths along the mountainside.