The distance is increasing at ?

Question

A plane flying with a constant speed of 24 km/min passes over a ground radar station at an altitude of 5 km and climbs at an angle of 50 degrees. At what rate is the distance from the plane to the radar station increasing 4 minutes later?

Answer 1

You can use cosine law to figure out the distance from the plane to the radar station

D² = (h)² + (v_pt)² - 2hv_pt*cos(θ) where h is initial height, v_p is the plane velocity, and θ is the angle between the two sides of the triangle (θ = 90 + 50 = 140)

In order to find the rate of the distance change with time at = 4 min, we can differentiate in time:

2D dD/dt = 2v_p²t - 2hv_pcos(θ)

dD/dt = (1/(2D))(2v_p²t - hv_pcos(θ)

Note that D is just the square root of the 1st equation: the RHS is all constants or calculable at t = 4.

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