Question

The cables on either side of a pedestrian

suspension bridge are in the shape of a

parabola. The towers that support the

cables are 100 feet apart and 27 feet

high. The cables are at a height of 2 feet

midway between the towers. What is the

height of a cable at a point that is 20 feet

from the center of the bridge?

Answer 1

6 ft

Explanation:

Since the shape of the cables on the bridge are to open up, the standard equation of the parabola produced is given as:

(x - h)² = 4p(y - k)

Where (h, k) is the vertex and focus is at (h, k+p)

From the question, the point (0, 2) is the vertex and point (50, 27) lie on the parabola. Hence:

(x - 0)² = 4p(y - 2)

x² = 4p(y - 2).

Sinc the tower is 100 ft apart and 27 ft height, hence the point 100/2 = 50 ft and 27 ft lie on the parabola

To find p, use (50, 27)

50² = 4p(27 - 2)

2500 = 4p(25)

100p = 2500

p = 25

hence:

x² = 4(25)(y - 2)

x² = 100(y - 2)

At a point of 20 feet (i.e x = 20), y is the height of the cable, hence:

20²=100(y-2)

400 = 100y - 200

100y = 600

y = 6

The height is 6 ft at a point of 20 ft