Question

Cliff jumping in Acapulco is considered very dangerous because the rock at a height of 35m does not overhang: you have to jump off forcefully in order to overcome a horizontal distance of approx. 8m before landing in the water. The jump path is almost parabolic in shape. The jumper reaches exactly 8m horizontally. Determine the equation of the quadratic function that can be used to describe the jump path if the vertex is placed in the coordinate origin.

Use the formula f(x) = ax^2 + bx + c

Answer 1

The quadratic function that models the jump path of the cliff diver in Acapulco is f(x) = (-35/64)x^2, with the path being parabolic and the vertex located at the origin of the coordinate system.

Step-by-step explanation:

The cliff jumping question involves determining the equation of a quadratic function (parabola) that represents the jump path of a diver.

Since the path is parabolic and the vertex is at the origin, we know the jumper must clear a horizontal distance of 8 meters with a maximum height of 35 meters.

The standard form of a quadratic function is f(x) = ax^2 + bx + c. Since the vertex is at the origin, this simplifies to f(x) = ax^2 because b and c are zero.

To find a, we use the fact that at the horizontal distance of 8m the height is -35m (because the jumper has descended 35m).

Substituting these values into the equation gives us -35 = a(8)^2, which simplifies to -35 = 64a. Solving for a gives us a = -35/64. Thus, the quadratic function describing the jump path is f(x) = (-35/64)x^2.

Answer 2

The equation of the quadratic function describing the jump path with the vertex at the origin is f(x) = -35/64 x2. The a value in the function was determined by using the known horizontal distance of 8m and the descent of 35m in the flight of the jump.

Step-by-step explanation:

To find the equation of the quadratic function describing the cliff jump path when the vertex is at the origin and the path is parabolic, we use the standard form f(x) = ax2 + bx + c. Since the vertex is at the origin, this implies that c is 0. Also, because it is symmetric about the y-axis, the linear term b will be 0. The equation reduces to f(x) = ax2.

To determine a, we know that the jumper covers a horizontal distance of 8m (the x-coordinate) and reaches a height of 35m before descending back to the water level (the y-coordinate). At this point (x=8), y will be -35, because we started at the highest point of the jump (the vertex at the origin) and descended 35 meters.

Substituting the known values into the equation gives us -35 = a(8)2 which simplifies to -35 = 64a. To find a, we divide both sides by 64, resulting in a = -35/64 or approximately -0.546875. Therefore, the equation describing the jump path is f(x) = -35/64 x2.