A man is 200 m south of a signal light. He uses a pair of binoculars to observe a car moving at 25 m/s perpendicularly toward the light. At what rate is the man rotating the binoculars at the instant the car is 45 m away from the signal?

Question

anonymous · Answer

<-45m->  <__CAR
     O________
 /\  |             /     <----25m/s
  |   | /
  |   |         /
  |   |       /
200|     /
  |   |   /
  |   |  /
  |   | 0/ Theta
 \/  |/
       MAN

amistre64 · Answer

the tangent of the angle is changing at a certain rate

anonymous · Answer

What?

amistre64 · Answer

tan(t) = d/200 right?

amistre64 · Answer

d is changing at a rate of 25 m.sec; 

200 isnt changing at all ....

anonymous · Answer

yeah thats right

amistre64 · Answer

tan(theta) = d/200  ; lets derive this with respect to time

(d(theta)/dt) sec^2(theta) = (1/200)  d(d)/dt ; now we solve for d(theta)/dt

amistre64 · Answer

$$\frac{d(\theta)}{dt}=\frac{25}{200*sec^2(\theta)}$$

amistre64 · Answer

we just need to determine the value of theta at the time indicated and fill it into our equation

amistre64 · Answer

tan(theta) = 45/200

theta = tan-1(45/200)

amistre64 · Answer

another method would be just to determine the value of sec(theta) and square it

amistre64 · Answer

the calculator tells me sec^2(theta) = 1681/1600

so we calculate:  25/(200*(1681/1600)) = 200/1681

amistre64 · Answer

or abt .11898

amistre64 · Answer

we got any options to judge by?

anonymous · Answer

AMAZING!!!

anonymous · Answer

that was right! (:

amistre64 · Answer

yay!! :)

anonymous · Answer

:D