Chapter 4.1, Problem 34E

Solution

To identify: The missing information. A central angle ? intercepts arcs $S_1$ and $S_2$ on two concentric circles with radii $r_1$ and $r_2$ , respectively.

  $\begin{array}{ccccc}\theta & r_1& s_1& r_2& s_2\\ ?& \text{8}\text{km}& \text{36km}& ?& \text{72km}\end{array}$

The lengths $r_2$ where lengths $S_2$ of the intercepted arc $S_2\text{= 72}$ kilometer and $\theta =4.5$ radian is 16 kilometers.

  $\begin{array}{ccccc}\theta & r_1& s_1& r_2& s_2\\ 4.5\text{radians}& \text{8}\text{km}& \text{36km}& 16\text{km}& \text{72km}\end{array}$

Given information:

The given length $S_1$ of the intercepted arc $S_1=36$ kilometer and radius $r_1=11$ kilometer.

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,

  $\begin{array}{l}\theta =\frac{36}{8}\\ \theta =4.5\end{array}$

Hence, the central angle $\theta$ when length $S_1$ of the intercepted arc $S_1=36$ kilometer and radius $r_1=8$ kilometer is $4.5$ radian.

Next to determine the radius $r_2$ when $S_2$ of the intercepted arc $S_2\text{= 72}$ kilometer and $\theta =4.5$ radian.

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

Thus,

  $\begin{array}{c}{r}_2=\frac{s_2}{\theta }\\ r_2=\frac{72}{4.5}\\ r_2=16\end{array}$

Hence, the lengths $r_2$ where lengths $S_2$ of the intercepted arc $S_2\text{= 72}$ kilometer and $\theta =4.5$ radian is 16 kilometers.