A spotlight is mounted at the top of a vertical pole that is 6 m tall
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A spotlight is mounted at the top of a vertical pole that is 6 m tall. A traffic cone that is 2 m tall is being pulled along a straight road directly away from the base of the pole at a constant rate of 0.40 m/s.
Let x(t) denote the horizontal distance (in metres) between the cone and the base of the pole at time t seconds, and let s(t) denote the length (in metres) of the cone’s shadow on the road at the same instant.
(a) Derive an explicit algebraic relationship among x(t), s(t), and the two fixed heights in this setting. Clearly state any assumptions you use about the geometry of the situation.
(b) Differentiate the relationship found in part (a) with respect to t to obtain an equation linking dx/dt and ds/dt. Confirm that all rates appearing in your equation have appropriate units.
(c) Suppose at a particular instant the shadow is getting longer at a rate of 0.15 m/s. Explain, without performing additional numerical calculations, whether the tip of the shadow (the farthest point from the pole receiving light) is moving faster or slower than the cone itself at that instant. Justify your conclusion verbally and symbolically.