Answer to Question #135761 in Calculus for Nicole Roxin Zambas

Question

A cable hangs in a parabolic arc between two columns 100 feet apart. The columns are 40 feet high and the lowest point on the suspended cable is 10 feet above the ground.
a. Find the equation of the arc if the vertex is the lowest point of the cable.
b. Find the height of the cable from the ground at a point 30 feet from the lowest point of the cable.

Answer 1

let f(x)= kx² + h₀

equation of the sought parabola where f(x) is the height from the ground, x the horizontal distance from the lower point of the parabola, h₀=10 is the height of the lower point of the parabola above the ground

for any of the columns, due to the symmetry of the parabola, we have

x = 50 f(x)=40

40 = k*50²+10

k = (40-10)/2500 = 3/250

equation of the sought parabola

f(x) = $\frac{3}{250}x^2+10$

for x= 30

f(x) = $\frac{3}{250}30^2+10 = 20.8$

Answer: $f(x)=\frac{3}{250}x^2+10;f(30) =20.8$