Question

At a given moment, a plane passes directly above a radar station at an altitude of
6 km. The plane's speed is 500 km/h.

How fast is the distance between the plane and the station changing half an hour later?

(Assume the plane maintains this speed and altitude over the half hour. Round your answer to three decimal places.)

Answer 1

Distance, D, between the plane and the radar station:

D^2 = y^2 + x^2

y = 6 km

x = Vt

V = constant = 500 km/h

D^2 = 6^2 + (500t)^2

D^2 = 36 + 250000t^2

Rate of change = d[D] /dt = D(t) ' = D'

[D^2]' = (2D)D' = 2*250000t    .... I used implicit derivative

=> D' = [2*250000t] /2D = [250000t] / D = [250000t] / [√(36 + 250000t^2) ]

t = 0.5 h => D' = [250000*0.5] / [√36 + 250000(0.5)^2] = 125000 / 250.07199

D' = 499.856 km / h

Answer: 499.856 km / h