Thread: Related Rates

Hey guys, any help would be appreciated.

A week ago, you were on duty in the state traffic patrol helicopter, hovering 1000 feet above Route 287, when a small car zipped by at a high speed. You aimed your radar gun and fired. The radar reported that at the moment you fired, the car was 2000 feet from your helicopter and moving away from you at a rate of 90ft/sec. You radioed ahead to your partner on the ground and had him stop the car and issue a speeding ticket.

Today's the car's driver has appeared in court. She tells the judge, "Your honor, I was clocked at 90 ft/sec. The speed limit is 65mph, which works out to a little over 95 ft/sec. No way was I speeding!" The judge turns to you and grunts, "Well?"

Nail this speeder. Show the judge how fast she was going.

Hello, pham07!

A week ago, you were on duty in the state traffic patrol helicopter,
hovering 1000 feet above Route 287, when a small car zipped by at a high speed.
You aimed your radar gun and fired. The radar reported that at the moment you fired,
the car was 2000 feet from your helicopter and moving away from you at a rate of 90 ft/sec.
You radioed ahead to your partner on the ground and had him stop the car and issue a speeding ticket.

Today's the car's driver has appeared in court.
She tells the judge, "Your honor, I was clocked at 90 ft/sec.
The speed limit is 65 mph, which works out to a little over 95 ft/sec.
No way was I speeding!" The judge turns to you and grunts, "Well?"

Nail this speeder. Show the judge how fast she was going.

Code:

    H *
      |  *
      |     *   s
 1000 |        *
      |           *
      |              *
      * - - - - - - - - *
      A        x        C

The helicopter is at $\displaystyle H.$
Its altitude is 1000 ft: .$\displaystyle HA = 1000$

The car is at $\displaystyle C.$
. . $\displaystyle s = HC$ is the distance from the helicopter to the car.
Let $\displaystyle x = AC$
. . The car's speed is: .$\displaystyle \frac{dx}{dt}$ feet per second.


In the right triangle: .$\displaystyle x^2 + 1000^2 \:=\:s^2$

Differentiate with respect to time: .$\displaystyle 2x\,\frac{dx}{dt} \:=\:2s\,\frac{ds}{dt}$
. . We have: .$\displaystyle \frac{dx}{dt} \:=\:\frac{s}{x}\,\frac{ds}{dt}$ .[1]

At the moment in question: .$\displaystyle s = 2000,\:\frac{ds}{dt} = 90$
. . and we find that: .$\displaystyle x \:=\:1000\sqrt{3}$

Substitute into [1]: .$\displaystyle \frac{dx}{dt} \:=\:\frac{2000}{1000\sqrt{3}}\,(90) \:=\:\frac{180}{\sqrt{3}} \;\approx\;104\text{ ft/sec}$

And $\displaystyle 104\text{ ft/sec} \;\approx\;71\text{ mph.}$

. . Gotcha!