Question

Speed Trap A highway patrol airplane flies 3 mi above a level, straight road at a constant rate of 120 mph. The pilot sees an oncoming car and with radar determines that at the instant the line-of-sight distance from plane to car is 5 mi the linstant the distance is decreasing at the rate of 160 \(\mathrm{mph}\) . Find the car's speed along the highway.

Short answer

The speed of the car is -190 mph in the opposite direction to the plane.

Step-by-step solution

Visualize the problem and assign variables to the quantities

Create a right triangle with the plane, the car, and the point on the ground vertically under the plane. Let's denote the distance from the plane to the car as \(c\), the distance from the plane to the point on the ground as \(a = 3 miles\) (that’s given as the plane's height), and denote the distance from the car to this point on the ground as \(b\). The plane's speed along the vertical line is 120 mph (miles per hour) and it's given that \(dc/dt = -160 mph\).

Create an equation relating the variables

We need an equation that relates \(a\), \(b\) and \(c\). Since these are sides of a right triangle, we can use Pythagoras' theorem \(a^2 + b^2 = c^2\).

Differentiate the equation with respect to time

Differentiate the Pythagorean equation with respect to time \(t\), we get \(2a·da/dt + 2b·db/dt = 2c·dc/dt\). Use the given data for the current moment of time: \(a = 3 miles\), \(c = 5 miles\), \(da/dt = 120 mph\), \(dc/dt = -160 mph\). Substituting these values into the differentiated equation, we get \(2·3·120 + 2b·db/dt = 2·5·(-160)\).

Find the rate in question

Finally, solve this equation for \(db/dt\), which represents the speed of the car: \(db/dt = (2·5·(- 160) - 2·3·120) / (2b) = -760/b mph.\nAt the current moment of time, \(b = \sqrt{c^2 - a^2} = \sqrt{5^2 - 3^2} = 4\), so the car's speed is \(db/dt = -760/4 = -190 mph\). The distance from the car to the point under the plane is decreasing, therefore, the car's speed is 190 mph in the opposite direction to the direction of the plane’s movement.