Question

A girl 5 feet tall buys a jump rope. At the ropes lowest point her hands will be 3 feet apart and 4 feet above the ground. With the rope taking shape of a parabola how long must the rope be? Solve the following.

Vertex:

Is it max/or min:

Focus:

Directix:

Equation:

Answer 1

The rope’s lowest point is the vertex of the parabola. It can be safely assumed to be the origin, i.e. the point (0,0). Further, as the rope touches its lowest point at the vertex, hence in this position, the length of the segment of the parabola is maximum.

The parabola passes through the points(-3/2,4), (0,0) and (3/2,4). Let the equation of the parabola be y = ax². On substituting x = 3/2 and y = 4 in this equation, we get 4 = a(3/2)² = 9a/4 so that a = 16/9. Thus, the equation of the parabola is y = (16/9)x² .

We know that the standard form of the equation of a parabola is is (x - h)² = 4p (y - k), where the focus is (h, k + p). Further, the directrix of the parabola is the line x = h-p.Here, h = 0, k = 0 and p = 9/64. Thus, the focus of the parabola is (0,9/64). The directrix is the line x =-9/64.