Answer:
To find the point where the paths of the sailboat and the speedboat intersect, we need to solve the system of equations formed by the given equations:
y = -x^2 + 4x + 3 (equation for the sailboat's path)
y = x + 3 (equation for the speedboat's path)
To find the x-coordinate of the intersection point, we can set the y-values of the two equations equal to each other:
-x^2 + 4x + 3 = x + 3
Now, let's solve this equation step by step:
First, let's move all the terms to one side to get a quadratic equation:
-x^2 + 4x + 3 - x - 3 = 0
Simplifying further:
-x^2 + 3x = 0
To solve this equation, we can factor out an x:
x(-x + 3) = 0
From this, we can see that either x = 0 or -x + 3 = 0. Solving each equation:
For x = 0, substituting this value into the equation for the sailboat's path:
y = -(0)^2 + 4(0) + 3
y = 3
So one point of intersection is (0, 3).
For -x + 3 = 0, solving for x:
-x = -3
x = 3
Substituting this value into the equation for the sailboat's path:
y = -(3)^2 + 4(3) + 3
y = -9 + 12 + 3
y = 6
So another point of intersection is (3, 6).
Therefore, the paths of the sailboat and the speedboat cross at two points: (0, 3) and (3, 6).
Explanation:
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