Related Rates Problem...Check My Work?

[msinc210]

Is this right?

An aircraft is flying at a constant altitude with a constant speed of 600 m/h. An antiaircraft missile is fired on a straight line perpendicular to the flight path of the aircraft so that it will hit the aircraft at a point P. At the instant the aircraft is 2 mi from the impact point P the missile is 4 m from P and flying at 1200 m/h. At that instant how rapidly is the distance between the missile and the aircraft decreasing?

Define missile path as x, define aircraft path as y. Distance between missile and aircraft is D.

D = rad(x^2 + y^2)
dD/dt = 1/2*(x^2 + y^2)^-1/2 *[2x(dx/dt) + 2y(dy/dt)]
plug in 4 for x, 2 for y
dD/dt = 1/2*(1/rad(20))*[8(dx/dt) + 4*(dy/dt)]
plug in -1200 for dx/dt and -600 for dy/dt
dD/dt = .11*(-9600+-2400)
dD/dt = -1320 m/h

Thanks!

 

[DrPhil]

Is this right?

An aircraft is flying at a constant altitude with a constant speed of 600 m/h. An antiaircraft missile is fired on a straight line perpendicular to the flight path of the aircraft so that it will hit the aircraft at a point P. At the instant the aircraft is 2 mi from the impact point P the missile is 4 m from P and flying at 1200 m/h. At that instant how rapidly is the distance between the missile and the aircraft decreasing?

Define missile path as x, define aircraft path as y. Distance between missile and aircraft is D.

D = rad(x^2 + y^2)
dD/dt = 1/2*(x^2 + y^2)^-1/2 *[2x(dx/dt) + 2y(dy/dt)]
plug in 4 for x, 2 for y
dD/dt = 1/2*(1/rad(20))*[8(dx/dt) + 4*(dy/dt)]
plug in -1200 for dx/dt and -600 for dy/dt
dD/dt = .11*(-9600+-2400)
dD/dt = -1320 m/h

Thanks!

\(\displaystyle dD/dt = [x(dx/dt) + y(dy/dt)]/D \)

...........\(\displaystyle = [4(-1200) + 2(-600)] / \sqrt{4^2 + 2^2}\)

...........\(\displaystyle = \frac{-6000}{4.472} = -1342\)

You rounded the square root a bit too soon.