Question

Two boats depart from a port located at (–10, 0) 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 (0, 5), and the second boat follows a path that can be modeled by a linear function and passes through the point (10, 4). At what point, besides the common starting location of the port, do the paths of the two boats cross?
(–6, 0.8)
(–6, 3.2)
(6, 3.2)
(6, 0.8)

Answer 1

Let the parabola be y = a x^2 + 5 (vertex at (0,5)). Passing through (−10,0) gives 0 = a(100) + 5 ⇒ a = −1/20, so y = −(1/20)x^2 + 5.

The line through (−10,0) and (10,4) has slope (4−0)/(10−(−10)) = 4/20 = 1/5, so y = (1/5)x + 2.

Solve −(1/20)x^2 + 5 = (1/5)x + 2:
Multiply by 20: −x^2 + 100 = 4x + 40 ⇒ x^2 + 4x − 60 = 0 ⇒ x = 6 or x = −10.

The other intersection (besides the port at −10) is x = 6, y = (1/5)(6) + 2 = 3.2.

Answer: (6, 3.2).