Related Rates and derivative Homework

Question

An air traffic controller spots 2 planes at the same altitude. They are traveling towards the control tower on paths perpendicular to each other. One plane is 150 km from the control tower and traveling at 450 km/h and the other plane is 200 km from the control tower and is traveling at 600 km/h

a) At what rate is the distance between the planes decreasing after 5 minutes have passed?

Answer 1

5 minutes = 5/60= 1/12 hour

plane x goes 600/12 = 50 km in 5 minutes

plane y goes 450/12 = 37.5 km in 5 minutes

200-50= 150 km away from tower for plane y the faster plane

150-37.5 = 112.5 km away for plane x, the slower plane

distance between the planes is the hypotenuse of a right triangle, h

h^2 = 150^2 + 112.5^2

h = 187.5 km after 6 minutes

hh' = xx' + yy' x'=400, y'=600, x=112.5, y=150

plug in the values and solve for h' = rate of change of the distance between the planes in 6 minutes

h'= [(112.5)(400)+150(600)]/187.5= 720 km/hr