7. An F-22 aircraft is flying at 500mph with an elevate on of 10,000ft  on a straight–line path that will take it directly over an anti–aircraft gun. How fast must the gun be able to turn to accurately track the aircraft when the plane is:

  (a) 1mileaway?
  (b) 1/5mileaway?
  (c) Directlyoverhead?

Here, A represents the F-22 aircraft, and G represents the anti-aircraft gun. The angle between the horizontal and the line from G to A is θ, which is the angle that the gun needs to turn in order to track the aircraft.

We can use the following formula to relate the rate at which the angle changes (dθ/dt) to the speed at which the aircraft is moving (v):

dθ/dt = v * sin(θ) / (R + h)

# Define the variables
v <- 500  # mph
h <- 10000  # ft
R <- 3960  # miles
# Calculate the values for (a)
dist_a <- sqrt(R^2 + 1^2)
theta_a <- atan(1/h)
dtheta_dt_a <- v * sin(theta_a) / (R + h)
# Calculate the values for (b)
dist_b <- sqrt(R^2 + (1/5)^2)
theta_b <- atan(1/(5*h))
dtheta_dt_b <- v * sin(theta_b) / (R + 1/5)
# Calculate the values for (c)
theta_c <- pi/2  # 90 degrees
dtheta_dt_c <- v / (R + h)
# Print the results
cat("For (a):\n")
## For (a):
cat("Distance = ", dist_a, " miles\n")
## Distance =  3960  miles
cat("Angle = ", theta_a, " radians\n")
## Angle =  1e-04  radians
cat("dθ/dt = ", dtheta_dt_a, " radians per second\n\n")
## dθ/dt =  3.581662e-06  radians per second
cat("For (b):\n")
## For (b):
cat("Distance = ", dist_b, " miles\n")
## Distance =  3960  miles
cat("Angle = ", theta_b, " radians\n")
## Angle =  2e-05  radians
cat("dθ/dt = ", dtheta_dt_b, " radians per second\n\n")
## dθ/dt =  2.525125e-06  radians per second
cat("For (c):\n")
## For (c):
cat("dθ/dt = ", dtheta_dt_c, " radians per second\n")
## dθ/dt =  0.03581662  radians per second