The center of an ice rink is located at (0, 0) on a coordinate system measured in meters. Susan is skating along a path that can be modeled by the equation y = 6x – x2 – 5. Luke starts at (10, –21) and skates along a path that can be modeled by a quadratic function with a vertex at (8, –9). If the rink is a circle with a radius of 35 meters, which statement best interprets the solution(s) of a system of equations modeling the paths of the skaters? A. The skaters’ paths intersect once, but that point is outside the rink.

B. The skaters’ paths intersect once, and that point is inside the rink.

C. The skaters’ paths intersect twice, but only one of those points is inside the rink.

D. The skaters’ paths intersect twice, and both points are inside the rink.

Option C:The skaters path intersect twice, but only one of those points is inside the rink.

Step-by-step explanation:

Coordinates of center of ice rink: (0,0) Radius of ice rink circle = 35 m

We know that the equation of a circle with center coordinates (a,b) and radius of (r) is given as; (x - a)² + (y - b)² = r²

Thus, the total area on which the skaters are skating will be given by the equation;

x² + y² = 35²

Now, we are told the path along which Susan is skating is modeled by the equation: y = 6x - x² - 5

While Luke starts at (10, –21) and skates along a path that can be modeled by a quadratic function with a vertex at (8, –9)

From equation of a parabola, the path along which like is skating can be modeled by;

(x - 8)² = 4a(y + 9)

To find a, we will substitute the coordinate started at to get;

(10 - 8)² = 4a(-21 + 9)

4 = 4a × -12

Divide both sides by 4 to get;

a = -1/12

Thus;

(x - 8)² = 4(-1/12)(y + 9)

(x - 8)² = (-1/3)(y + 9)

This gives: 3(x - 8)² = -(y + 9)

I've drawn the graph on demos and attached it.

From the graph we can see that the skaters points intersect twice but one is inside the rink circle while the other is outside the rink circle. The point at which they intersect outside the rink was slightly cropped out because of size. But it is clearly seen that they are both approaching point of intersection.

Thus, correct answer is Option C.

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