Answer to Question #246835 in Analytic Geometry for Nooanmeee

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 the suspended cable is 5 feet above the ground. Find the equation of the arc if the vertex is the lowest point of the cable.

Also, find the height of the cable at a point 10 feet from one of the poles.

Answer 1

Suppose that the equation of the parabola is $y(x)=ax^2+c, a>0.$

The lowest point on the suspended cable is 5 feet above the ground

$$y(0)=a(0)^2+c=5=>c=5$$

The poles are 100 feet apart. The poles are 30 feet high.

$$y(-50)=y(50)=30$$

Then

$$30=a(50)^2+5=>a=\dfrac{35}{2500}=0.014$$

a) The equation of the arc is

$$y(x)=0.014x^2+5$$

b)

$$y(40)=0.014(40)^2+5=27.4$$

$$y(60)=0.014(60)^2+5=55.4$$

The height of the cable at a point 10 feet from one of the poles is 27.4 feet or 55.4 feet.