Let's assume the rocket rises vertically and its speed is 500 ft/s when it has risen 3000 ft.

Question

A television camera is positioned 4000 ft from the base of a rocket launching pad. The angle of elevation of the camera has to change at the correct rate in order to keep the rocket in sight. Also, the mechanism for focusing the camera has to take into account the increasing distance from the camera to the rising rocket. Let's assume the rocket rises vertically and its speed is 500 ft/s when it has risen 3000 ft.



(a) How fast is the distance from the television camera to the rocket changing at that moment?

(b) If the television camera is always kept aimed at the rocket, how fast is the camera's angle of elevation changing at that same moment?

Answer 1

The first step is to write down the relevant equations.

For part a we care about the distance from the rocket to the camera: this is the hypotenuse

We use the Pythagorean theorem for this.

a² + b² = c² a = 3000 b = 4000 c = 5000

Because we care about the rate of change we must take the derivative with respect to time of this equation.

When taking the derivative of variables that depend on time follow the chain rule.

d/dt (a²) = 2a * da/dt (2a is the derivative of the outside da/dt is the derivative of the inside)

Solving the full equation we get.

2a * (da/dt) + 2b * (db/dt) = 2c * (dc/dt) Remember da/dt is just a variable

db/dt = 0 because it doesn't change over time.

Simplifying we get.

a * da/dt = c * dc/dt

Plugging in values we finally get.

3000 * 500 = 5000 * dc/dt

Solving we get dc/dt = 300 ft/s

For part b the equation we start with is the tangent of an angle formula.

tan(θ) = a/b I am using a and b from the previous part.

Take the derivative of both sides, and note that θ and a depend on time while b is a constant

sec²(θ) * dθ/dt = (da/dt) /b

Plugging in values sec(θ) = 1/cos(θ) = 1/(adjacent/hypotenuse) = 5000/4000 = 1.25

(1.25)² * dθ/dt = 500 / 4000

Solve for dθ/dt

dθ/dt = 0.08 radians per second

or

dθ/dt = 4.58 degrees per second when the rocket is 3000 ft above the pad