Calculus - A plane flying with a constant speed of 4 km/min passes over a ground radar station at

A plane flying with a constant speed of 4 km/min passes over a ground radar station at an altitude of 4 km and climbs at an angle of 25 degrees. At what rate, in km/min, is the distance from the plane to the radar station increasing 4 minutes later?

Rate = ????? km/min

Draw a obtuse angled triangle ABC, with angle
B is the position of radar station

A is the position of plane at t = 0

C is the position of plane at t = 4

AB = 4 km

BC is the distance between plane and radar station after 4 min

let BC = s

let AC = = p

dp/dt = 4 km/min

using law of cosines

s^2 = p^2 + 4^2 - 2p*(4)cos(115)

s^2 = p^2 + 16 + 3.384p-------(1)

when t = 4, p = 4*4 = 16 km,

=> s^2 = 256 + 16 + 54.144

==> s^2 = 326.144

s = 18.06 km

now differentiate eqn (1) with respect to t

2s ds/dt = 2p dp/dt + 3.384 dp/dt

s ds/dt = dp/dt (p + 1.692)

substitute s = 18.06, p = 16 and dp/dt = 4

18.06 ds/dt = 4(17.692)

ds/dt = 2*(17.692)/9

= 3.93 km/min