To find the area of sector LMN in circle M, we can use the formula for the area of a sector:
\[ \text{Area of sector} = \frac{\theta}{360^\circ} \times \pi r^2 \]
where \(\theta\) is the angle in degrees and \(r\) is the radius of the circle.
In this case, we have the following information:
- \( \theta = 36^\circ \)
- The length of \( LM = 5 \) (this represents the radius of the circle, so \( r = 5 \)).
Now, plug the values into the formula:
\[ \text{Area of sector} = \frac{36}{360} \times \pi \times (5^2) \]
Calculating the area step-by-step:
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Calculate \( \frac{36}{360} \): \[ \frac{36}{360} = \frac{1}{10} = 0.1 \]
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Calculate \( 5^2 \): \[ 5^2 = 25 \]
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Combine these results in the formula: \[ \text{Area of sector} = 0.1 \times \pi \times 25 \] \[ \text{Area of sector} = 2.5\pi \]
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Use \( \pi \approx 3.14159 \) to find the area: \[ \text{Area of sector} \approx 2.5 \times 3.14159 \approx 7.854 \]
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Finally, round to the nearest hundredth: \[ \text{Area of sector} \approx 7.85 \]
Thus, the area of sector LMN is approximately \( \boxed{7.85} \).