A plane flying with a constant speed of 180 kmh passes over

Solution

The point 1 km above the station, the station itself and the plane form a triangle. I will define the following. D = distance from station to plane h = distance to the point above the station = 1 km s = distance from the point above the station to the plane We want s at 1 minute later. Speed of plane = 180 km/h s = 180*(1/60) = 3 km ..... 60 minutes in an hour We also want D at this time. The angle at the point above the station = 90 + 30 = 120 degrees Law of cosines D^2 = 1^2 + 3^2 - 2*1*3*cos(120) = 1 + 9 + 3 = 13 D = SQRT(13) So the sides of the triangle are 1, 3 and SQRT(13) at 1 minute after the plane passes over the radar station. Write the law of cosines only using s and D and not actual values. D^2 = 1 + s^2 - 2*1*s*cos(120) D^2 = 1 + s^2 + s Now take the derivative. 2*D*(dD/dt) = (2*s + 1)*(ds/dt) Solve this for dD/dt. dD/dt = [(2*s + 1)/(2*D)]*(ds/dt) ds/dt = speed of the plane = 50 m/s We can use s and D in km since their units cancel each other out. dD/dt = [(2*3 + 1)/(2*SQRT(13))]*50 m/s dD/dT = (175/13)SQRT(13) m/s = 48.5 m/s