Answer to Question #246835 in Analytic Geometry for Nooanmeee

Question #246835

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.


1
Expert's answer
2021-10-06T14:06:22-0400

Suppose that the equation of the parabola is y(x)=ax2+c,a>0.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=5y(0)=a(0)^2+c=5=>c=5

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


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

Then


30=a(50)2+5=>a=352500=0.01430=a(50)^2+5=>a=\dfrac{35}{2500}=0.014

a) The equation of the arc is


y(x)=0.014x2+5y(x)=0.014x^2+5

b)


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




y(60)=0.014(60)2+5=55.4y(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.


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