A drone takes off from point A and flies horizontally toward the east while…
MATHEMATICS · Multi-part · AP · AP Precalculus · Functions with Parameters, Vectors, and Matrices · Functions with Parameters, Vectors, and Matrices
Problem
A drone takes off from point A and flies horizontally toward the east while simultaneously gaining altitude. At time t = 0 s, the drone is directly above A at ground level. Its horizontal speed remains 4 m/s, and its rate of climb is 2 m/s. At the same instant, a delivery truck located 30 m east of A begins to drive due south on a straight road at 3 m/s, remaining on the ground (z = 0).
Assume a three–dimensional coordinate system with origin at A, x-axis pointing east, y-axis pointing north, and z-axis pointing upward.
(a) Express the position of the drone, rD(t), and the position of the truck, rT(t), as vector functions of time, stating each component explicitly.
(b) At t = 5 s, write the relative position vector of the drone with respect to the truck.
(c) Without explicitly differentiating the distance between the two objects, determine whether the distance is increasing or decreasing at t = 5 s. Justify your conclusion by referencing both the relative position and the relative velocity vectors, and include appropriate units in your reasoning.