Question

consider circle y with radius \\(3\text{ m}\\) and central angle \\(xyz\\) measuring \\(70^\circ\\).

what is the approximate length of minor arc \\(xz\\)?
round to the nearest tenth of a meter.

\\(1.8\text{ meters}\\)
\\(3.7\text{ meters}\\)
\\(15.2\text{ meters}\\)
\\(18.8\text{ meters}\\)

Explanation:

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<reasoning>

Identify given values and formula

\[

$$\begin{aligned} &r = 3\text{ m} \\ &\theta = 70^\circ \\ &s = \frac{\theta}{360^\circ} \cdot 2\pi r \end{aligned}$$

\]

Calculate the arc length

\[

$$\begin{aligned} &s = \frac{70}{360} \cdot 2 \cdot \pi \cdot 3 \\ &s = \frac{7}{36} \cdot 6\pi = \frac{7\pi}{6} \\ &s \approx \frac{7 \cdot 3.14159}{6} \approx 3.665\text{ m} \end{aligned}$$

\]

Round to the nearest tenth

\[
s \approx 3.7\text{ meters}
\]
</reasoning>

<answer>
<mcq-option>1.8 meters</mcq-option>
<mcq-correct>3.7 meters</mcq-correct>
<mcq-option>15.2 meters</mcq-option>
<mcq-option>18.8 meters</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify given values and formula

\[

$$\begin{aligned} &r = 3\text{ m} \\ &\theta = 70^\circ \\ &s = \frac{\theta}{360^\circ} \cdot 2\pi r \end{aligned}$$

\]

Calculate the arc length

\[

$$\begin{aligned} &s = \frac{70}{360} \cdot 2 \cdot \pi \cdot 3 \\ &s = \frac{7}{36} \cdot 6\pi = \frac{7\pi}{6} \\ &s \approx \frac{7 \cdot 3.14159}{6} \approx 3.665\text{ m} \end{aligned}$$

\]

Round to the nearest tenth

\[
s \approx 3.7\text{ meters}
\]
</reasoning>

<answer>
<mcq-option>1.8 meters</mcq-option>
<mcq-correct>3.7 meters</mcq-correct>
<mcq-option>15.2 meters</mcq-option>
<mcq-option>18.8 meters</mcq-option>
</answer>

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