Question

The current San Francisco cable railway is driven by two large 14-foot-diameter drive wheels, called sheaves. Because of the figure-eight system used, the cable subtends a central angle of 270°on each sheave. Find the length of cable riding on one of the drive sheaves at any given time.

Answer 1

The length of the cable is around 32.97 feet.

Explanation:

The problem is about a circular sector, where the angle is 270° and the lenght of the subtended arc is the arc is the unknown variable.

The radius of the drive wheel is

`r=(d)/(2)=(14)/(2)=7ft`

The central angle is 270°.

The length of the arc is

`L=(2\pi r \theta)/(360\°)`

Replacing all values, we have

`L=(2(3.14)(7)(270\°))/(360\°)=(11869.2)/(360) \approx 32.97`

Thereofre, the length of the cable is around 32.97 feet.

Answer 2

The length of an arc is given by (θ/360)2πr, where θ is the angle subtended by the arc at the center and r is the radius.
The angle θ in this case is 270°, while the radius is 7 feet, diameter is 14 feet
Thus the length of the of the cable will be;
=(270/360)× 22/7×14
= 0.75 × 44
= 33 feet