Chapter 4.1, Problem 98E

To determine

Find the measurement of the radius of the circle.

Answer to Problem 98E

  $1.39\text{inches}$

Explanation of Solution

Given:

The Central angle $\text{θ}=330{}^0$ and length of the arc on the circle $\text{s}=8\text{inches}$

Concept Used:

The radian measure, $\text{θ}$ of a central angle is defined as the ration of the length of the arc the angle subtends, $\text{s}$ divided by the radius of the circle, $\text{r}$

$\begin{array}{l}\text{θ}=\frac{\text{length}\text{of}\text{subtended}\text{arc}}{\text{length}\text{of}\text{radius}}=\frac{\text{s}}{\text{r}}\\ \\ \text{r}=\frac{\text{s}}{\text{θ}}\end{array}$

Calculation:

First convert $\text{θ}=330{}^0$ into radian measure.

  $\begin{array}{l}{180}^0=\text{π}\text{radian}\\ 1^0=\frac{\text{π}}{180}\text{radian}\\ {330}^0=330·\frac{\text{π}}{180}\text{radian}=\frac{\text{11π}}{6}\text{radian}\end{array}$

Given: The central angle: $\text{θ}=\frac{\text{11π}}{6}\text{radian}$ and the length of arc on the circle $\text{s}=8\text{inches}$

We know that:

  $\begin{array}{l}\text{θ}=\frac{\text{length}\text{of}\text{subtended}\text{arc}}{\text{length}\text{of}\text{radius}}=\frac{\text{s}}{\text{r}}\\ \\ \text{r}=\frac{\text{s}}{\text{θ}}\end{array}$

  $\text{r}=\frac{\text{s}}{\text{θ}}=\frac{\text{8}}{\frac{\text{11π}}{6}}\text{inches}=8·\frac{\text{6}}{\text{11π}}\text{inches}=\frac{\text{48}}{\text{11π}}\text{inches}=\frac{\text{48}}{11·\text{3}\text{.14}}\text{inches}=1.39\text{inches}[\text{π}=3.14]$

Thus, the measurement of the length of the radius is $1.39\text{inches}$