A car travels west along a straight road that coincides with the positive –axis on a…
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A car travels west along a straight road that coincides with the positive x–axis on a coordinate grid. At time t=0 s the car is 200 ft east of the origin and moves at a constant speed of 25 ft s−1 toward the origin. Simultaneously, a weather balloon is released from the origin and ascends vertically (in the z–direction) at a constant rate of 15 ft s−1. Throughout the motion the balloon remains directly above the origin and the car remains on the x–axis.
(a) Derive a formula for the distance R(t) between the car and the balloon for t≥0. Clearly indicate which coordinates you assign to each object.
(b) Obtain an expression for dtdR in terms of t. Simplify the expression as much as algebraically convenient.
(c) Compute dtdR at the instant when the balloon is 180 ft above the ground. State units and indicate whether the distance is increasing or decreasing at that instant.