the angle between 2 ships?

Question

Ship A is traveling west toward a light house at 15 km/hr. Ship B is traveling north away from lighthouse at 10km/hr. x is distnace between ship A and lighthouse at time t, and y is distance between ship B and lighthouse at time t.

Let theta be the angle with the vertex of ship A. find the rate of change of theta in radians per hour, when x=4km and y=3km.


THANKS!!!

i got the same answer as my friend, but i dont know how to cancel the km^2...to just leave rad/hr.
i took the derivative of tan(theta)=y/x, and plugged in the values i have (x=4, y=3, dx/dt= - 15 km/hr, dy/dt= 10 km/hr, and i solved to find theta (.6435 rad of 36.869degrees).
am i doing this right?

well my answr is 3.4 rad/hr.
i dnt kno how u got that answer?
im confused...:(

Answer 1

Let f = y/x and g = arctan(f)

g' = 1/(1+y/x)^2 * (dy/dt /x - dx/dt y/x^2)

g' = 1/(1+3/4)^2 * (10/4 + 15*3/16) = 1.7347 rad/hr
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What's your formulation for d(theta)/dt???
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Ooops...Put the squared in the wrong place.
g' = 1/(1+(3/4)^2) * (10/4 + 15 *3/16) = 3.4 rad/hr
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1/(1+(3km/4km)^2)) has no units
10/4 has km/hr/km = /hr
15km/hr * 3km/16 km^2 = /hr
radians has no units

Answer 2

I believe the answer is (pi/5). Sorry dont know the shortcut for the pi symbol on the keyboard. Theta at A in a 3-4-5 triangle is (pi/5). Theta at the rates 10 and 15 are (pi/5). Theta at A after 1 hour (3+10 and 4+15) is (pi/5) as well as at 2 hours.

To get pi/5 take the inverse tangent of (3/4) or (10/15) or (13/19). You will get a degree answer, convert that to radians with the conversion factor (pi/180). The answers are just slightly off from pi/5