Mathematics
Mathematics
Problem
A helicopter flies due north at a constant altitude of 800 m above ground level. At time t = 0 s, it is located 1 km south of a highway intersection, moving north at 60 m/s. Simultaneously, a truck is 1.2 km west of the same intersection and travels east toward the intersection at 22 m/s, remaining on the ground. Let the origin be the intersection point, the x-axis point east, the y-axis point north, and the z-axis point upward. (a) Express the squared distance between the helicopter and the truck as a function of time. (b) Derive an expression for the rate at which the distance between the helicopter and the truck is changing at any time t. (c) Evaluate the rate found in part (b) at t = 10 s, and interpret its sign in the context of the physical situation. (d) Determine the time (if any) after t = 0 s at which the distance between the helicopter and the truck is minimized. Justify your answer analytically.Intuitive explanation
This is a related rates problem about finding how fast the distance between two moving objects changes. Set up position functions for both the helicopter and truck using their starting locations and velocities, then use the distance formula to express the separation between them as a function of time. The key is to differentiate that distance with respect to time—you don't need to find the actual distance, just how it's changing at a specific moment.
Intuitive explanation
This is a related rates problem about finding how fast the distance between two moving objects changes. Set up position functions for both the helicopter and truck using their starting locations and velocities, then use the distance formula to express the separation between them as a function of time. The key is to differentiate that distance with respect to time—you don't need to find the actual distance, just how it's changing at a specific moment.