5
QuizMultiple Choice

Modeling with Systems — Quiz

Question 5 • Advanced Algebra: Concepts and Connections

A lighthouse is located at open paren 1 comma 2 close paren in a coordinate system measured in miles. A sailboat starts at open paren negative 7 comma 8 close paren and sails in a positive x -direction along a path that can be modeled by a quadratic function with a vertex at open paren 2 comma negative 6 close paren Which system of equations can be used to determine whether the boat comes within 5 miles of the lighthouse?

Answer
A
open paren x minus 1 close paren squared plus open paren y minus 2 close paren squared is equal to 25 y is equal to negative 14 over 81 times open paren x plus 7 close paren squared plus 8
B
open paren x minus 1 close paren squared plus open paren y minus 2 close paren squared is equal to 5 y is equal to negative 14 over 81 times open paren x plus 7 close paren squared plus 8
C
open paren x minus 1 close paren squared plus open paren y minus 2 close paren squared is equal to 25 y is equal to 14 over 81 times open paren x minus 2 close paren squared minus 6
D
open paren x minus 1 close paren squared plus open paren y minus 2 close paren squared is equal to 5 y is equal to 14 over 81 times open paren x minus 2 close paren squared minus 6

Explanation

The points within 5 miles of the lighthouse form a circle centered at (1, 2), so its equation is (x − 1)² + (y − 2)² = 5² = 25. The boat’s path has vertex (2, −6), so write it as y = a(x − 2)² − 6. Substituting the starting point (−7, 8) gives 8 = 81a − 6, so a = 14/81. Thus the system in choice A is correct.

AI-written and checked against the verified answer.

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