Wien's Law gives us a way to find the peak wavelength (\(\lambda_{max}\)) emitted by a black body in relation to its temperature (T) in Kelvin. The formula for Wien’s Law is:
\[ \lambda_{max} = \frac{b}{T} \]
where \(b\) is Wien's displacement constant, approximately equal to \(2898 , \mu m \cdot K\) (micrometers times Kelvin).
First, if we don't have the temperature from Question #8, we would not be able to calculate the exact peak wavelength. However, to convert our answer into nanometers, we note that:
1 micrometer (µm) = 1000 nanometers (nm).
So, we can express Wien’s Law in terms of nanometers (noting that 1 µm = 1000 nm):
\[ \lambda_{max} = \frac{2898 , \mu m \cdot K}{T} \times 1000 , nm/\mu m \]
\[ \lambda_{max} = \frac{2898000 , nm \cdot K}{T} \]
To calculate the peak wavelength, you would substitute the temperature of Betelgeuse (if given in question #8) into the equation.
Assuming we applied this to a hypothetical temperature—let's say Betelgeuse's effective temperature is around 3500 K (which is a rough estimate). The calculation would be:
\[ \lambda_{max} = \frac{2898000 , nm \cdot K}{3500 , K} \approx 828 nm \]
So based on the options you provided—and assuming Betelgeuse's temperature used in your question is indeed close to this—828 nm would be the selected answer.
Finally, please substitute the actual temperature if it differs from this assumed value for precise calculations.
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