Question

chapter ii.8
score: 3/6 answered: 4/6
question 5
there are 111 zuma - radians in a circle.
(a) determine the zuma - radian measure of angle that is $\frac{165}{296}$ radians.
(b) determine the zuma - radian measure of angle that is 78 degrees.
(c) determine the degree measure of an angle that is 119 zuma - radians.
(d) determine the radian measure of an angle that is 187 zuma - radians.

Explanation:

Step1: Establish conversion factor

Since there are 111 Zuma - radians in a full - circle (i.e., $2\pi$ radians or 360 degrees), the conversion factor from regular radians to Zuma - radians is $\frac{111}{2\pi}$, and from Zuma - radians to regular radians is $\frac{2\pi}{111}$, and from Zuma - radians to degrees is $\frac{360}{111}$, and from degrees to Zuma - radians is $\frac{111}{360}$.

Step2: Solve part (a)

To convert $\frac{165}{296}$ radians to Zuma - radians, we multiply by the conversion factor $\frac{111}{2\pi}$.
\[

$$\begin{align*} \text{Zuma - radians}&=\frac{165}{296}\times\frac{111}{2\pi}\\ &=\frac{165\times111}{296\times2\pi}\\ &=\frac{18315}{592\pi}\approx\frac{18315}{592\times3.14159}\\ &=\frac{18315}{1860.82128}\approx9.84 \end{align*}$$

\]

Step3: Solve part (b)

First, convert 78 degrees to radians: $78\times\frac{\pi}{180}=\frac{13\pi}{30}$ radians. Then convert to Zuma - radians: $\frac{13\pi}{30}\times\frac{111}{2\pi}=\frac{13\times111}{30\times2}=\frac{1443}{60} = 24.05$

Step4: Solve part (c)

To convert 119 Zuma - radians to degrees, we use the conversion factor $\frac{360}{111}$.
\[
\text{Degrees}=119\times\frac{360}{111}=\frac{119\times360}{111}=\frac{42840}{111}\approx386
\]

Step5: Solve part (d)

To convert 187 Zuma - radians to radians, we use the conversion factor $\frac{2\pi}{111}$.
\[
\text{Radians}=187\times\frac{2\pi}{111}=\frac{374\pi}{111}\approx\frac{374\times3.14159}{111}=\frac{1175.95466}{111}\approx10.6
\]

Answer:

(a) $\frac{18315}{592\pi}\approx9.84$
(b) 24.05
(c) $\frac{42840}{111}\approx386$
(d) $\frac{374\pi}{111}\approx10.6$