Even though the description of the path sounds confusing,
a parabola with a focus of (8,16) and the directrix at y=-8 is indeed
y = (1/48)(x-8)^2 + 4
I first did y=1/2(x-h)^2+k
p=2y
y=1/48(x-8)^2+4
Is this correct? Thank you for checking my work
a parabola with a focus of (8,16) and the directrix at y=-8 is indeed
y = (1/48)(x-8)^2 + 4
The standard equation of a parabola with the vertex at (h, k) and the focus (h, k + p) is:
(y - k) = (1 / (4p))(x - h)^2
In this case, the vertex is at (8, 4) and the focus is at (8, 16), which means the distance from the vertex to the focus is p = 16 - 4 = 12.
Plugging in the values into the equation, we get:
(y - 4) = (1 / (4 * 12))(x - 8)^2
Simplifying,
(y - 4) = (1 / 48)(x - 8)^2
So the correct equation of the parabola is:
y = (1 / 48)(x - 8)^2 + 4
Therefore, your answer is almost correct, but the sign in front of 48 should be positive instead of negative.
(y - k) = (1 / (4p))(x - h)^2
Where (h, k) represents the coordinates of the vertex, and p represents the distance between the vertex and the focus. In this case, the focus is located at (8, 16), so h = 8 and k = 16.
To find the value of p, we need to consider that the parabola is equidistant from the floor and the pivot point at the ceiling. Since the rope is attached at the ceiling, which is 8 feet above the starting point, the distance between the directrix and the vertex should also be 8 feet.
The equation of the directrix is y = -8, so the point on the parabola directly above the directrix should be at a height of 8 units from the vertex.
Since the parabola opens upwards, the distance between the vertex and the point on the parabola above the directrix is p. Thus, p = 8.
Plugging these values into the standard form equation, we get:
(y - 16) = (1 / (4 * 8))(x - 8)^2
Simplifying this, we have:
(y - 16) = (1 / 32)(x - 8)^2
Multiplying through by 32 to eliminate the fraction:
32(y - 16) = (x - 8)^2
Expanding the right side:
32y - 512 = x^2 - 16x + 64
Rearranging and combining like terms:
x^2 - 16x + 32y = 576 - 512
x^2 - 16x + 32y = 64
So, the equation of the parabola that the flying trapeze artist swings along is: x^2 - 16x + 32y = 64.