Asked by nb
A lighthouse is located at (1, 2) in a coordinate system measured in miles. A sailboat starts at (–7, 8) and sails in a positive x-direction along a path that can be modeled by a quadratic function with a vertex at (2, –6). Which system of equations can be used to determine whether the boat comes within 5 miles of the lighthouse?
StartLayout Enlarged Left-Brace 1st Row (x minus 1) squared + (y minus 2) squared = 5 2nd Row y = StartFraction 14 Over 81 EndFraction (x minus 2) squared minus 6 EndLayout
StartLayout Enlarged Left-Brace 1st Row (x minus 1) squared + (y minus 2) squared = 25 2nd Row y = StartFraction 14 Over 81 EndFraction (x minus 2) squared minus 6 EndLayout
StartLayout Enlarged Left-Brace 1st Row (x minus 1) squared + (y minus 2) squared = 5 2nd Row y = Negative StartFraction 14 Over 81 EndFraction (x + 7) squared + 8 EndLayout
StartLayout Enlarged Left-Brace 1st Row (x minus 1) squared + (y minus 2) squared = 25 2nd Row y = Negative StartFraction 14 Over 81 EndFraction (x + 7) squared + 8 EndLayout
All Answers 1
Answered by
GPT-5 mini
AI
The correct system is the second one:
(x − 1)^2 + (y − 2)^2 = 25
y = (14/81)(x − 2)^2 − 6
Reason: "Within 5 miles" gives a circle of radius 5: (x−1)^2+(y−2)^2 = 5^2 = 25. The parabola has vertex (2,−6), so y = a(x−2)^2 − 6; using the point (−7,8) gives 8 = a(−9)^2 − 6 ⇒ a = 14/81.
(x − 1)^2 + (y − 2)^2 = 25
y = (14/81)(x − 2)^2 − 6
Reason: "Within 5 miles" gives a circle of radius 5: (x−1)^2+(y−2)^2 = 5^2 = 25. The parabola has vertex (2,−6), so y = a(x−2)^2 − 6; using the point (−7,8) gives 8 = a(−9)^2 − 6 ⇒ a = 14/81.
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