A pair of fire towers uses technology to search for forest fires. Let the center of the forest be the origin on a coordinate grid. The first fire tower is located 5 km east and 8 km south of the center, and the second fire tower is located 7 km west and 4 km north of the center. The technology at the first tower can locate fires within a radius of 10 km and the second tower within a radius of 9 km. The system of inequalities represents this scenario. open parenthesis x minus 5 close parenthesis squared plus open parenthesis y plus 8 close parenthesis squared is less than or equal to 100; open parenthesis x plus 7 close parenthesis squared plus open parenthesis y minus 4 close parenthesis squared is less than or equal to 81 At which location, from the origin, would a fire be detected by both towers?
Answer
A
1 km west and 2 km south
B
2 km west and 1 km north
C
2 km east and 3 km south
D
3 km east and 2 km north
Explanation
“1 km west and 2 km south” has coordinates (−1, −2). Its squared distance from the first tower at (5, −8) is (−1 − 5)² + (−2 + 8)² = 72, which is at most 100. Its squared distance from the second tower at (−7, 4) is (−1 + 7)² + (−2 − 4)² = 72, which is at most 81. So this location is within range of both towers.
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