A sedan tall drives straight away from a streetlamp that is tall

MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

A sedan 1.8 m1.8\,\text{m} tall drives straight away from a streetlamp that is 12 m12\,\text{m} tall, both situated on horizontal pavement. Let x(t)x(t) be the distance (in meters) between the car and the base of the streetlamp at time tt seconds, y(t)y(t) be the length (in meters) of the car's shadow on the pavement, and v=dxdt=22 m/sv = \frac{dx}{dt} = 22\,\text{m/s} be the constant speed of the car. Assume the head of the car's driver remains at a fixed height equal to the car's roof. (1) Part 1: (a) Derive an equation that relates xx and yy at any time tt. (2) Part 2: (b) Differentiate the equation from part (a) with respect to time to obtain a relation between dydt\frac{dy}{dt} and dxdt\frac{dx}{dt}. (3) Part 3: (c) Determine the instantaneous speed, in meters per second, of the tip of the car’s shadow (the point on the pavement farthest from the lamp that is still in shadow) when the car is 30 m30\,\text{m} from the lamp.

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