Question

A sailboat with a sick passenger aboard is following a parabolic path given by the equation y=(−x)^2 +2x+8. The boat's captain sent out a distress signal. A speedboat is trying to catch the sailboat to provide medical aid and is on a path given by the linear equation y=−x+8. Besides the y-intercept (starting location), where do the paths of the two boats cross?

a) (0, 8)
b) (4, 4)
c) (2, 6)
d) (6, 2)

Answer 1

The paths of the sailboat and the speedboat cross at the point (0, 8).

Step-by-step explanation:

The paths of the sailboat and the speedboat will cross at a point where their y-coordinates are equal. To find this point, we can set the equations for the y-values of both paths equal to each other and solve for x:

(-x)^2 + 2x + 8 = -x + 8

Simplifying the equation, we get x^2 + 3x = 0.

Factoring out x, we have x(x + 3) = 0.

So x = 0 or x = -3.

Substituting these values into one of the equations, we find that the y-values are both 8.

Therefore, the paths of the two boats cross at the point (0, 8), so the answer is (a) (0, 8).