Answer to Question #136417 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 Oy is between the columns, Ox is the ground.

a. Parabolic arc is symmetric about Oy, so the equation is

$y=ax^2+0x+c,$

or $y=ax^2+c.$

Let's take points $(0, 10), (50, 40),$ substitute them into the equation:

$\begin{cases} 10=a\cdot0^2+c, \\ 40=a\cdot50^2+c; \end{cases}$ $\Rarr$ $\begin{cases} c=10, \\ a=\frac{3}{250}. \end{cases}$

So the equation is $y=\frac{3}{250}x^2+10.$

b. The height of the cable from the ground at a point 30 feet from the lowest point of the

cable is $\frac{3}{250}30^2+10=\frac{270}{25}+10=10.8+10=20.8$(ft).

Answer: $y=\frac{3}{250}x^2+10$, $\medspace20.8$ ft.