(Solved): 3. A cop sits in a parked car 10 feet from a straight road. As you drive along the road, the cop ai ...

3. A cop sits in a parked car 10 feet from a straight road. As you drive along the road, the cop aims a radar gun at your car. Let $s$ be the distance from your car to the cop in feet. The radar gun measures the rate at which your distance from the cop is changing with respect to time, which is $ds/dt$. Let $x$ be the distance, in feet, of your car from the point on the road that is closest to the cop, so your car's velocity is $dx/dt$. (a) Draw and label a diagram of this scenario, and write $\frac{dx}{dt}$ in terms of $\frac{ds}{dt}$, $s$, and $x$. (b) Use part (a) to explain why $?\frac{ds}{dt}?<?\frac{dx}{dt}?$. Thus the radar gun's measurement of $?\frac{ds}{dt}?$ always underestimates your car's speed. (This is why if the radar gun measures a speed greater than the speed limit on the road, the driver gets a ticket.) (c) Does the conclusion in part (b) depend on the cop's car being 10 feet, rather than some other (positive) distance, from the road?