Question

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?

Answer 1

We are given:

the center of the rink at the origin.
A skating path (Susan): y = 6x - x^2 - 5
Starting point of Luke = (10, -21) 
Path (Luke) = quadratic eq'n with vertex at (8, -9)
Radius = 35 meters 


The solution that best interprets the path of the skaters is to substitute Luke's starting point to Susan's path or set-up a quadratic equation with vertex of (8,-9) and then equate to Susan's path to solve for their intersection. 

Answer 2

Solution:

The center of ice rink having center (0,0) and radius = 35 meters,  that is total area on which skaters are skating is given by

x²+y²= 35²  [ Equation of circle having center (a,b) and radius r is given by  (x-a)²+ (y-b)²=r²]

Susan path is modeled by the equation:

$y = 6x - x^2 - 5.$

Path of Luke is parabolic path

As vertex of parabola is (8,-9) and it passes through or starting point of Luke is (10,-21).

Equation of parabola having vertex at (8,-9) and slanting towards negative y  axis is given by:

(x-8)²=4 a (y+9)

It passes through (10,-21).

(10-8)²= 4 a (-21+9)

4 = 4 a× (-12)

a= $\frac{-1}{12}$

So, equation of parabola becomes , after substituting value of a

3(x-8)²= -(y+9)

Drawing these graphs on desmos graphing , and getting the point of intersection of these curves

The solution of the system of equation i.e path of skaters  is the point of intersection of equation of circle and two parabolas both slanting towards negative y axis.