A missile is rising vertically from a point on the ground 75,000 feet from a tracking station. The missile is rising at a rate of 16500 feet per minute at the instant when it is 38,000 feet high.What is the rate of change, in radians per minute, of the missile's angle of elevation from the tracking station at this instant?

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.