The Federal Aviation Administration (FAA) requires that airplanes flying at the same altitude maintain a distance of at least

Question

feet. An air traffic controller monitors a display of airplane locations in a coordinate grid. She observes two planes, labeled
and
in the following figure, flying at the same altitude. Each unit on the display represents
feet.

The figure presents two data points in a grid, with the origin labeled O. The numbers 1 through 7, in increments of 1, are indicated on the horizontal axis. The numbers negative 2 through 3, in increments of 1, are indicated on the vertical axis. Data point A has coordinates 7 comma 2, and data point B has coordinates 2 comma negative 2.
Should the air traffic controller alert the planes that they are flying too close?

Responses

Yes; the planes are approximately
feet apart.

Answer A: Yes; the planes are approximately 6,400 feet apart.
A

Yes; the planes are approximately
feet apart.

Answer B: Yes; the planes are approximately 12,800 feet apart.
B

No; the planes are approximately
feet apart.

Answer C: No; the planes are approximately 16,100 feet apart.
C

No; the planes are approximately
feet apart.

Answer 1

To determine the distance between the two planes (points A and B) given their coordinates A(7, 2) and B(2, -2), we can use the distance formula:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Substituting the coordinates into the distance formula:

• \(x_1 = 7\), \(y_1 = 2\) (for point A)
• \(x_2 = 2\), \(y_2 = -2\) (for point B)

We calculate:

\[ d = \sqrt{(2 - 7)^2 + (-2 - 2)^2} \] \[ = \sqrt{(-5)^2 + (-4)^2} \] \[ = \sqrt{25 + 16} \] \[ = \sqrt{41} \]

Now, since each unit on the grid represents a certain number of feet, we need to convert \(\sqrt{41}\) to feet by multiplying it by the scaling factor (the number of feet represented by each unit).

Assuming each unit represents \(x\) feet, the distance in feet would be:

\[ d = \sqrt{41} \times x , \text{feet} \]

Given the numerical values in the multiple-choice answers, let's analyze potential values for \(x\):

1. Assuming \(x = 100\) feet/unit:

   • \(\sqrt{41} \approx 6.4\)
   • Hence, the distance is approximately \(6.4 \times 100 = 640\) feet.

2. Assuming \(x = 200\) feet/unit:

   • Distance is \(6.4 \times 200 = 1280\) feet.

3. Assuming \(x = 400\) feet/unit:

   • Distance is \(6.4 \times 400 = 2560\) feet.

4. Assuming \(x = 500\) feet/unit:

   • Distance is \(6.4 \times 500 = 3200\) feet.

This suggests that unless we know the exact conversion of units to feet, we should compare the distances obtained with the multiple choices.

Given the options:

• Approximately 6,400 feet (potentially 100 ft/unit)
• Approximately 12,800 feet (potentially 200 ft/unit)
• Approximately 16,100 feet (potentially unlikely based on typical unit sizes)

To conclude, the distance calculated needs to be less than the FAA requirement to say they are too close.

• If each unit represents 200 feet: The distance would be approximately \(12,800\) feet, which would rise to an alert.

• If each unit is 100 feet: the approximate distance would be \(6400\) feet, but still a potential alert.

Ultimately, there’s no option for a unit conversion yielding large distances, so check the scaling factor to confirm compliance with the FAA.

If this scenario refers to real-world context on appropriate separation rules, typically 3 nautical miles (~18,000 ft) is adherence.

Based on the plausible answer choices and configuration, if the assumption holds true at 200 feet per unit, the answer would be likely B: Yes; the planes are approximately 12,800 feet apart.