A spy tracks a rocket through a telescope to determine its velocity. The rocket is traveling vertically from a launching pad located 10 km away. At a certain moment, the spy's instruments show that the angle between the telescope and the ground is equal to 60 degrees and is changing at a rate of 0.5 radians/min. What is the rocket's velocity at that moment?
To determine the rocket's velocity, we can use trigonometry and related rates. Let's break down the problem:
1. We have a right triangle formed by the telescope, the ground, and the rocket's trajectory.
2. The spy's position can be considered as the vertex of the right angle in the triangle.
3. The distance between the launching pad and the spy is the base of the right triangle, which is given as 10 km.
4. The angle between the telescope and the ground is changing at a rate of 0.5 radians/min.
To find the rocket's velocity, we need to find the rate of change of the height of the rocket, or the rate at which the distance between the rocket and the spy is changing. We can use the concept of related rates to find this.
Let's denote the height of the rocket as h (in km) and the angle as θ (in radians). The rate of change of the height of the rocket with respect to time (dh/dt) is what we're looking for.
Now, we can use the tangent function to relate the angle and the height of the rocket:
tan(θ) = h / 10
Differentiating implicitly with respect to time (t), we get:
sec²(θ) * dθ/dt = (dh/dt) / 10
We are given that dθ/dt = 0.5 radians/min, and we're looking for (dh/dt) when θ = 60 degrees.
First, we need to convert θ from degrees to radians:
θ (in radians) = (π/180) * 60°
Substituting the known values into the equation, we have:
sec²(θ) * (0.5 radians/min) = (dh/dt) / 10
We can simplify sec²(θ) by using the identity: sec²(θ) = 1 + tan²(θ).
Plugging in the value of θ in radians:
sec²(π/3) * (0.5 radians/min) = (dh/dt) / 10
Using the fact that sec(π/3) = 2, we have:
2² * (0.5 radians/min) = (dh/dt) / 10
4 * 0.5 = (dh/dt) / 10
2 = (dh/dt) / 10
Now, we can solve for (dh/dt) by multiplying both sides of the equation by 10:
20 = dh/dt
Therefore, at that moment, the rocket was moving at a velocity of 20 km/min.