Question

A cable hangs in a parabolic arc between two poles 100 feet apart. The poles are 30 feet high and the lowest point on a suspended cable is 5 feet above the ground

a) find the equation of the arc if the vertex is the lowest points of the cable

b) find the height of the cable at a point 10 feet from one of the poles

Answer 1

a) Let point $(0, 5)$ be the vertex of the parabola. Then the equation of the parabola will be

$$y=ax^2+5, a>0$$

Given $y(-50)=y(50)=30.$ Substitute

$$a(50)^2+5=30$$

$$a=0.01$$

The equation of the arc is

$$y=0.01x^2+5$$

b)

If $x_1=50-10=40$

$$y=0.01(40)^2+5$$

$$y=21$$

The height of the cable at a point 10 feet from one of the poles is $21$ feet.