Answer: To find the rate at which the distance from the plane to the radar station is increasing a minute later, we can use the concept of related rates and trigonometry.
Let:
D = distance from the plane to the radar station (the hypotenuse of the right triangle formed by the plane, the radar station, and the ground)
h = altitude of the plane above the radar station (1 km)
θ = angle of elevation (30°)
v = speed of the plane (240 km/h)
The rate at which the distance D is changing can be found using the following formula:
Rate of change of D = (Rate of change of h) / sin(θ)
First, let's find the rate of change of h (the altitude of the plane) a minute later. Since the plane is flying at a constant speed and climbs at an angle of 30°, the vertical component of its speed (rate of change of h) can be found using trigonometry:
Rate of change of h = v * sin(θ)
Rate of change of h = 240 km/h * sin(30°)
Rate of change of h ≈ 240 km/h * 0.5
Rate of change of h ≈ 120 km/h
Now, we can find the rate at which the distance D is changing:
Rate of change of D = (Rate of change of h) / sin(θ)
Rate of change of D ≈ 120 km/h / sin(30°)
Using the value of sin(30°) ≈ 0.5:
Rate of change of D ≈ 120 km/h / 0.5
Rate of change of D ≈ 240 km/h
So, the distance from the plane to the radar station is increasing at a rate of approximately 240 km/h a minute later.