Question

Express the required measurement in exact form.

A bicycle wheel with radius 26" rotates through an arc that measures 80°. What is the length of the arc of the tire that touched the ground?

help me

Answer 1

r = 26

radians = 80*pi/180
= 4*pi/9
= 4pi/9

length = (26*4pi)/9
= 104pi/9

Answer 2

One full turn of a circle in radians is equal to \(2\pi\) radians. That is to say that the radius of the circle times the angle in radians which it moves through will give the length of the arc (i.e. \(C=2\pi r\)). Since 360 degrees = \(2\pi\) radians, then this will correspond to 2/9ths of a circle, or \(2\pi\times 2/9=4\pi/9\).

therefore the total arc length of the wheel in contact with the ground will be \[26\times\frac{4\pi}{9}=\frac{104}{9}\pi\]