## To find the speed of the rocket at a specific instant, we need to use related rates and calculus. Here's how we can approach this problem step by step:

Step 1: Write down the given values and equations:

- We are given that the camera is rotating at a rate of 3/169 radians per second (dTheta/dt = 3/169).

- The camera is located 4 km from the launch pad.

- At the specific instant we are interested in, the height of the rocket is 13 km.

Step 2: Determine the relationship between the variables:

We are given the relationship tangentTheta = h/4. This equation relates the angle of rotation of the camera (Theta) to the height of the rocket (h).

Step 3: Find the derivative of the equation:

Take the derivative of both sides of the equation with respect to time (t) to handle the related rates problem:

d/dt(tanTheta) = d/dt(h/4)

Using the chain rule, we get:

sec^2Theta * dTheta/dt = (1/4) * dh/dt

Step 4: Solve for dh/dt:

We need to solve for dh/dt, which represents the rate at which the height of the rocket is changing.

Rearranging the equation from step 3, we get:

dh/dt = (sec^2Theta * dTheta/dt) / (1/4)

Step 5: Substitute the given values:

To evaluate dh/dt, we need to substitute the given values into the equation. We know that h = 13 km, and we are given that dTheta/dt = 3/169 radians per second.

Step 6: Evaluate secTheta:

To find secTheta, we need to use the right triangle formed by the camera, launch pad, and rocket. From this triangle, we know that secTheta = sqrt(h^2 + 16)/h. Substituting the value of h = 13 km, we can calculate secTheta.

Step 7: Calculate dh/dt:

Substitute the values of secTheta, dTheta/dt, and (1/4) into the equation for dh/dt derived in step 4, and compute the value.

Step 8: Convert the answer to km/sec:

Since the speed of the rocket is measured in km/sec, make sure to convert the value of dh/dt to the appropriate units.

Following these steps should allow you to determine the speed of the rocket at the specific instant when its height is 13 km.