Two boats depart from a port located at open paren negative 10 comma 0 close paren in a coordinate system measured in kilometers, and they travel in a positive x -direction. The first boat follows a path that can be modeled by a quadratic function with a vertex at open paren 0 comma 5 close paren , and the second boat follows a path that can be modeled by a linear function and passes through the point open paren 10 comma 4 close paren At what point, besides the common starting location of the port, do the paths of the two boats cross?
Answer
A
open paren negative 6 comma 0 point 8 close paren
B
open paren 6 comma 3 point 2 close paren
C
open paren negative 6 comma 3 point 2 close paren
D
open paren 6 comma 0 point 8 close paren
Explanation
The quadratic path has the form y = ax² + 5 because its vertex is (0, 5). Using the starting point (−10, 0) gives a = −0.05. The linear path passes through (−10, 0) and (10, 4), so its slope is 0.2 and its equation is y = 0.2x + 2. Setting the paths equal gives −0.05x² + 5 = 0.2x + 2, with solutions x = −10 and x = 6. The second is the crossing besides the port; its y-coordinate is 0.2(6) + 2 = 3.2, so the point is (6, 3.2), choice B.
AI-written and checked against the verified answer.