Final answer:
The question involves using calculus and the Pythagorean theorem to find the rate at which the distance from the plane to the radar station is increasing, given the speed and climb angle of the plane. By differentiating the Pythagorean equation with respect to time and substituting the horizontal and vertical components of the plane's velocity, we can determine the rate of increase of the hypotenuse, which represents the distance to the radar.
Step-by-step explanation:
The question deals with the situation where a plane flies at a constant speed and altitude and then begins to climb at a 30-degree angle. To determine the rate at which the distance from the plane to the radar station is increasing, we can use calculus and the Pythagorean theorem in the context of right-angled triangles.
We know the plane flies at a constant speed of 300 km/hr and climbs at an angle of 30 degrees. Let's denote the altitude of the plane as y (in kilometers) and the horizontal distance it has traveled since passing the radar station as x. The distance from the radar to the plane is the hypotenuse of the triangle, which we'll call z. At the moment the plane passes above the radar, x is 0 and y is 1 km.
The relationship between x, y, and z is given by the Pythagorean theorem: z² = x² + y². To find the rate at which z is increasing, we need to differentiate both sides of this equation with respect to time t. Since the plane is climbing at a 30-degree angle, for every unit of altitude gained, two units of horizontal distance are covered (since tan(30 degrees) = 1/2).
Using the given speed of the plane (300 km/hr), the horizontal component of its velocity vx is 300 cos(30 degrees), and the vertical component vy is 300 sin(30 degrees). Now, differentiating the Pythagorean equation and substituting for vx and vy, we can solve for dz/dt, which is the rate at which z is increasing.