## I apologize for the confusion. Let's go through the calculations again.

For the first problem, we have the equation tan(theta) = h/2, where theta is the angle and h is the height of the rocket. We want to find dh/dt when theta = 1/200 radians/sec.

To find dh/dt, we'll take the derivative of both sides of the equation with respect to time (t):

sec^2(theta) * d(theta)/dt = (1/2) * (dh/dt)

d(theta)/dt is given as 1/200 radians/sec, so let's substitute that in:

sec^2(theta) * (1/200) = (1/2) * (dh/dt)

Now, we need to find the value of sec^2(theta), which is equal to 1 + tan^2(theta). Since tan(theta) = h/2, we can substitute that in as well:

sec^2(theta) = 1 + (h/2)^2

Now let's plug in the values:

1 + (h/2)^2 * (1/200) = (1/2) * (dh/dt)

The value of h is given as 20 km. Plugging that in, we get:

1 + (20/2)^2 * (1/200) = (1/2) * (dh/dt)

Simplifying further:

1 + 100/40000 = (1/2) * (dh/dt)

1 + 1/400 = (1/2) * (dh/dt)

801/400 = (1/2) * (dh/dt)

Now we can solve for dh/dt:

dh/dt = (2 * 801)/400 = 1602/400 = 4.005 km/sec (approx)

So the speed of the rocket at that instant is approximately 4.005 km/sec.

Now let's move on to the second problem.

We have the equation 16/x = 6/(x - D), where x is the distance from the lamp to the end of the shadow, and D is the distance from the lamp to the girl. We want to find dD/dt, the rate at which D is changing.

To solve for dD/dt, we'll take the derivative of both sides of the equation with respect to time (t):

d(16/x)/dt = d(6/(x - D))/dt

Let's simplify the equation first:

16/x = 6/(x - D)

Now let's differentiate both sides:

d(16/x)/dt = d(6/(x - D))/dt

-16/x^2 * dx/dt = -6/(x - D)^2 * (dx/dt - dD/dt)

We are given dx/dt as 5 ft/sec. Substituting that in:

-16/x^2 * 5 = -6/(x - D)^2 * (5 - dD/dt)

Simplifying further:

-80/x^2 = -6/(x - D)^2 * (5 - dD/dt)

Now let's solve for dD/dt:

dD/dt = 5 - (80/x^2) * (x - D)^2 / 6

Since we don't have information about x and D, we cannot find a specific numerical value for dD/dt. However, this equation gives us the general formula to calculate the rate at which the girl is walking, given the values of x and D.

I hope this explanation clarifies the process for solving these related rates problems. If you have further questions, feel free to ask!