n the figure below, a highway rest area (at D) and radar stations (at A and B) lie on a level east-west line; A is 9,000 feet due west of D. An airplane (at C) is shown directly above the rest area, flying due west at a constant speed of 300 feet per second and at a constant altitude of 12,000 feet. The airplane is located at a straight-line distance of 15,000 feet from the radar station at A and 13,000 feet from the radar station at B.Which of the following values is closest to the distance, in feet, between the 2 radar stations?
Question
n the figure below, a highway rest area (at D) and radar stations (at A and B) lie on a level east-west line; A is 9,000 feet due west of D. An airplane (at C) is shown directly above the rest area, flying due west at a constant speed of 300 feet per second and at a constant altitude of 12,000 feet. The airplane is located at a straight-line distance of 15,000 feet from the radar station at A and 13,000 feet from the radar station at B.Which of the following values is closest to the distance, in feet, between the 2 radar stations?
Solution
To find the distance between the two radar stations, we can use the Pythagorean theorem.
First, we need to find the distance from the airplane to the rest area (D). We know that the airplane is at an altitude of 12,000 feet and is at a straight-line distance of 15,000 feet from radar station A and 13,000 feet from radar station B.
We can use the Pythagorean theorem to find the distance from the airplane to the rest area (D). The theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
For radar station A, the hypotenuse is 15,000 feet and one side is 12,000 feet (the altitude of the airplane). So, the other side (AD) can be found using the equation:
(15,000)^2 - (12,000)^2 = (AD)^2 225,000,000 - 144,000,000 = (AD)^2 81,000,000 = (AD)^2 AD = sqrt(81,000,000) = 9,000 feet
Similarly, for radar station B, the hypotenuse is 13,000 feet. So, the other side (BD) can be found using the equation:
(13,000)^2 - (12,000)^2 = (BD)^2 169,000,000 - 144,000,000 = (BD)^2 25,000,000 = (BD)^2 BD = sqrt(25,000,000) = 5,000 feet
Therefore, the distance between the two radar stations (AB) is AD + BD = 9,000 feet + 5,000 feet = 14,000 feet.
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Back 41:53Finish LaterSubmitIn the figure below, a highway rest area (at D) and radar stations (at A and B) lie on a level east-west line; A is 9,000 feet due west of D. An airplane (at C) is shown directly above the rest area, flying due west at a constant speed of 300 feet per second and at a constant altitude of12,000 feet. The airplane is located at a straight-line distance of 15,000 feet from the radar station at A and 13,000 feet from the radar station at B.
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