Question

air traffic controllers are watching two planes on radar to ensure there is enough distance between them. plane a took off at 10:00 a.m., and plane b took off at the same runway 5 minutes later. both planes are flying at the same direction angle and the same path. at 10:10 a.m., the airport’s radar system detected plane a at (24, 18) and plane b at (8, 6). the scale on the radar is 1 unit = 25 miles. which vector represents the path from plane a to plane b, and what is the actual distance between them? components: (16,12), distance between planes: about 20 miles components: (16,12), distance between planes: about 500 miles components: (-16, -12), distance between planes: about 20 miles components: (-16, -12), distance between planes: about 500 miles

Explanation:

Step1: Find the vector components

The formula for the vector from point \(A(x_1,y_1)\) to point \(B(x_2,y_2)\) is \(\langle x_2 - x_1,y_2 - y_1
angle\).
Given \(A(24,18)\) and \(B(8,6)\), then \(x_2 - x_1=8 - 24=- 16\), \(y_2 - y_1=6 - 18=-12\). So the vector is \(\langle-16,-12
angle\).

Step2: Calculate the distance between the two points

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
Substitute \(x_1 = 24,y_1 = 18,x_2 = 8,y_2 = 6\) into the formula:
\(d=\sqrt{(8 - 24)^2+(6 - 18)^2}=\sqrt{(-16)^2+(-12)^2}=\sqrt{256 + 144}=\sqrt{400}=20\) units.

Step3: Convert the distance from units to miles

Since \(1\) unit \( = 25\) miles, the actual distance \(D=20\times25 = 500\) miles.

Answer:

components \(\langle-16,-12
angle\), distance between planes: about \(500\) miles.