To find the rate of change, in radians per minute, of the missile's angle of elevation from the tracking station, we can use trigonometry.
Let's define some variables:
- Let θ be the angle of elevation of the missile from the tracking station.
- Let x be the horizontal distance from the tracking station to the missile.
- Let y be the vertical distance from the tracking station to the missile.
From the given information, we have the following:
x = 75,000 feet
y = 38,000 feet
dy/dt = 16,500 feet/minute
We want to find dθ/dt, the rate of change of θ with respect to time. To find this, we need to relate the variables x, y, and θ using trigonometry.
tan(θ) = y/x
Differentiating both sides of this equation with respect to time t, we get:
sec^2(θ) * dθ/dt = (dy/dt * x - y * dx/dt) / x^2
Since we are looking for dθ/dt, we can rearrange the equation as follows:
dθ/dt = (dy/dt * x - y * dx/dt) / (x^2 * sec^2(θ))
Now we can substitute in the given values to find the rate of change:
x = 75,000 feet
y = 38,000 feet
dx/dt = 0 (since the missile is rising vertically)
dy/dt = 16,500 feet/minute
Next, we need to find the value of sec^2(θ) to substitute into the equation.
Using the Pythagorean theorem, we know that:
x^2 + y^2 = (75,000)^2
Solving for sec^2(θ), we get:
sec^2(θ) = 1 + (y^2 / x^2)
Now we can substitute all the values into the equation:
dθ/dt = (dy/dt * x - y * dx/dt) / (x^2 * sec^2(θ))
= (16,500 * 75,000 - 38,000 * 0) / (75,000^2 * (1 + (38,000^2 / 75,000^2)))
Simplifying the expression:
dθ/dt = (16,500 * 75,000) / (75,000^2 * (1 + (38,000^2 / 75,000^2)))
= 11 / (75,000 * (1 + (38,000^2 / 75,000^2)))
Calculating this expression yields the rate of change, in radians per minute, of the missile's angle of elevation from the tracking station at the given instant.