Chapter 4.1, Problem 75E

To determine

To explain: how the length of the intercepted arc change when the radius of a circle increases and the magnitude of a central angle is held constant.

Answer to Problem 75E

Arc length is proportional to the radius of circle.

Explanation of Solution

Concept Used:

For a circle of radius r , the arc length s of the sector with central angle $\theta$ (in radian measure) is:

  $s=r\theta$

For the variables x and y , the statement y varies directly as x can be translated as:

  $y=kx$

Where k is a non-zero constant called the constant of proportionality.

Now, the arc length of the circle is,

  $s=r\theta$

And here the magnitude of a central angle is held constant.

Then it satisfies above definition of being proportional.

Hence, it can be said that if the magnitude of a central angle is held constant, then the arc length is proportional to the radius of circle.