Chapter 4.1, Problem 33E

Solution

To identify: The missing information from the table.

$\theta$	$r_1$	$S_1$	$r_2$	$S_2$
$?$	11 cm	9 cm	44 cm	?

The lengths $S_2$ of the intercepted arc where radius $r_2=44$ centimeter and $\theta =0.82$ radian is 36 centimeters.

$\theta$	$r_1$	$S_1$	$r_2$	$S_2$
$\text{0}\text{.82}\text{R}\text{}\text{adian}$	11 cm	9 cm	44 cm	$\text{36}\text{cm}$

Given information:

A central angle ? intercepts arcs $S_1$ and $S_2$ on two concentric circles with radii $r_1$ and $r_2$ respectively such that”

$\theta$	$r_1$	$S_1$	$r_2$	$S_2$
$?$	11 cm	9 cm	44 cm	?

The given length $S_1$ of the intercepted arc $S_1=9$ centimeter and radius $r_1=11$ centimeter.

Explanation:

Consider the given information.

Recall that $S=r\theta$ where $r$ is the radius of the circle and $\theta$ is the central angle.

Thus,

  $\begin{array}{l}{S}_1=r_1\theta \\ \theta =\frac{s_1}{r_1}\end{array}$

So,

  $\begin{array}{l}\theta =\frac{9}{11}\\ \theta =0.82\end{array}$

Hence, the central angle $\theta$ when length $S_1$ of the intercepted arc $S_1=9$ centimeter and radius $r_1=11$ centimeter is $0.82$ radian.

Next to determine the lengths $S_2$ of the intercepted arc where radius $r_2=44$ centimeter and $\theta =0.82$ radian.

Recall that $S=r\theta$ where $r$ is the radius of the circle and $\theta$ is the central angle

Thus,

  $\begin{array}{l}{\text{S}}_{\text{2}}\text{= 0}\text{.82 × 44}\\ {\text{S}}_{\text{2}}\text{= 36}\text{centimeter}\end{array}$

Hence, the lengths $S_2$ of the intercepted arc where radius $r_2=44$ centimeter and $\theta =0.82$ radian is 36 centimeters.