Answer:
Horizontal distance = 0 m and 6 m
Step-by-step explanation:
Height of a rider in a roller coaster has been defined by the equation,
y = [tex]\frac{1}{3}x^{2}-2x+8[/tex]
Here x = rider's horizontal distance from the start of the ride
i). [tex]y=\frac{1}{3}x^{2}-2x+8[/tex]
[tex]=\frac{1}{3}(x^{2}-6x+24)[/tex]
[tex]=\frac{1}{3}[x^{2}-2(3x)+9-9+24][/tex]
[tex]=\frac{1}{3}[(x^{2}-2(3x)+9)+15][/tex]
[tex]=\frac{1}{3}[(x-3)^2+15][/tex]
[tex]=\frac{1}{3}(x-3)^2+5[/tex]
ii). Since, the parabolic graph for the given equation opens upwards,
Vertex of the parabola will be the lowest point of the rider on the roller coaster.
From the equation,
Vertex → (3, 5)
Therefore, minimum height of the rider will be the y-coordinate of the vertex.
Minimum height of the rider = 5 m
iii). If h = 8 m,
[tex]8=\frac{1}{3}(x-3)^2+5[/tex]
[tex]3=\frac{1}{3}(x-3)^2[/tex]
(x - 3)² = 9
x = 3 ± 3
x = 0, 6 m
Therefore, at 8 m height of the roller coaster, horizontal distance of the rider will be x = 0 and 6 m
