A commercial freight carrier is flying at a constant speed of 400mi/h and is traveling 4 mi east for every 3 mi north. A private plane is observed to be 210 mi due east to the commercial carrier, traveling 12 mi north for every for 5 mi west.

a.If it is known that the two planes are on collision course, how fast is the private plane flying ?
b. When will the collision take place if it is not averted?

To solve this problem, we can use the concept of relative velocity. The relative velocity of the private plane with respect to the commercial carrier can be calculated by subtracting the velocity of the commercial carrier from the velocity of the private plane.

Let's break down the given velocities and distances:

For the commercial carrier:
- Constant speed: 400 mi/h
- Traveling 4 mi east for every 3 mi north

For the private plane:
- Velocity in the east direction: 210 mi
- Velocity in the north direction: 12 mi
- Velocity in the west direction: 5 mi

Now let's calculate the relative velocity of the private plane with respect to the commercial carrier.

Step 1: Find the net velocity of the commercial carrier.
Since the carrier is moving 4 mi east for every 3 mi north, we can find its net velocity by finding the ratio of these distances.

Net velocity of the commercial carrier = (4 mi) / (3 mi) = 4/3 mi

Step 2: Calculate the relative velocity of the private plane with respect to the commercial carrier.
Since the private plane is observed to be 210 mi due east to the commercial carrier, it means that it is traveling 210 mi faster than the commercial carrier's net velocity.

Relative velocity = Velocity of the private plane - Net velocity of the commercial carrier
Relative velocity = 210 mi - 4/3 mi = (630 mi - 4 mi) / 3 mi = 626/3 mi

a. How fast is the private plane flying?
The private plane's speed can be calculated by taking the magnitude of the relative velocity.

Speed of the private plane = |Relative velocity| = |626/3 mi| ≈ 208.67 mi/h

Therefore, the private plane is flying at a speed of approximately 208.67 mi/h.

b. When will the collision take place if it is not averted?
To find the time of collision, we need to divide the distance between the two planes by the relative velocity.

Distance between the two planes = 210 mi (given)

Time of collision = Distance / Speed
Time of collision = 210 mi / 208.67 mi/h ≈ 1.005 hours

Therefore, the collision will take place approximately 1.005 hours after they were observed to be 210 mi apart.